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Plus Two Physics Notes Chapter 9 Ray Optics and Optical Instruments is part of Plus Two Physics Notes. Here we have given Plus Two Physics Notes Chapter 9 Ray Optics and Optical Instruments.

Kerala Plus Two Physics Notes Chapter 9 Ray Optics and Optical Instruments

Introduction
In this chapter, we consider the phenomena of reflection, refraction and dispersion of light, using the ray picture of light.

Reflection Of Light Byspherical Mirrors
Laws of reflection:

1. According to the first law of reflection, the angle of reflection equals the angle of incidence.
2. According to the second law of reflection, the incident ray, reflected ray and the normal to the point of incidence all lie in the same plane.

1. Sign convention:

• According to this convention, all distances are measured from the pole of the mirror or the optical centre of the lens.
• The distances measured in the same direction as the incident light are taken as positive and
  those measured in the direction opposite to the direction of incident light are taken as negative.
• The heights measured upwards are taken as positive. The heights measured downwards are taken as negative.

2. Focal length of spherical mirrors:
Reflection of light: Spherical mirrors are of two types.

• Concave mirror
• Convex mirror

Principal focus of a concave mirror:

A narrow parallel beam of light, parallel and close to the principal axis, after reflection converges to a fixed point on the principal axis is called principal focus of concave mirror.
Principal focus of a convex mirror:

A narrow parallel beam of light, parallel and close to the principal axis, after reflection appears to diverge from a point on the principal axis is called principal focus of convex mirror.
Relation connecting focal length and radius of curvature:

Consider a ray AB parallel to principal axis incident on a concave mirror at point B and is reflected along BF. The line CB is normal to the mirror as shown in the figure.
Let θ be angle of incidence and reflection.
Draw BD ⊥ CP,
In right angled ΔBCD,
Tanθ = \(\frac{B D}{C D}\) _____(1)
In right angled ΔBFD,
Tan2θ = \(\frac{B D}{F D}\) _____(2)
Dividing (1)and(2)
\(\frac{\tan 2 \theta}{\tan \theta}=\frac{C D}{F D}\) ____(3)
If θ is very small, then tanθ ≈ θ and tan2θ ≈ 2θ
The point B lies very close to P. Hence CD ≈ CP and FD ≈ FP From (3) we get

3. The mirror equation:

Let points P, F, C be pole, focus, and centre of curvature of a concave mirror. Object AB is placed on the principal axis. A ray from AB incident at E and then reflected through F. Another ray of light from B incident at pole P and then reflected. These two rays meet at M. The ray of light from point B is passed through C. Draw EN perpendicular to the principal axis.
ΔIMF and ΔENF are similar.
ie. \(\frac{I M}{N E}=\frac{I F}{N F}\) _____(1)
but IF = PI – PF and NF = PF (since aperture is small)
hence eq. (1) can be written as

[∵ NE = AB)
ΔABP and ΔIMP are similar

From eq.(2) and eq.(3), we get

applying sign convention we get
PI = -v
PF = -F
PA = -u
Substituting these values in eq.(4) we get

This is called mirror formula or mirror equation.
Linear magnification:
Linear magnification is defined as the ratio of the height of the image to the height of the object.

Consider an object AB having height h_o, which produces an image IM having height h_i
In the figure, ΔABP and ΔIMP are equal. ie.

Applying sign convention
PI = -V, PA = -u, h_i = -ve and h_o = +ve
We get

But we know \(\frac{h_{i}}{h_{0}}\) = m (magnification) ie.

This formulae is true fora concave mirror and convex mirror.
Relation connecting v, f, and m
We have
\(\frac{1}{u}+\frac{1}{v}=\frac{1}{f}\)
Multiplying throughout by ‘v’, we get
\(\frac{v}{u}+\frac{v}{v}=\frac{v}{f}\)
But m = -v/u
ie. -m + 1 = \(\frac{v}{f}\)
m = 1 – \(\frac{v}{f}\)

Relation connecting u, f and m
We know
\(\frac{1}{u}+\frac{1}{v}=\frac{1}{f}\)
Multiplying throughout by ‘u’ we get

Refraction
The phenomenon of bending of light when it travels from one medium to another is known as refraction.
Light from rarer to denser medium:

When light travels from a rarer medium to a denser medium, it deviates towards the normal.
Light from denser to rarer medium:

When light travels from a denser medium to a rarer medium, it deviates away from the normal.

Laws of refraction:
First law:
The incident ray, the refracted ray, and the normal at the point of incidence are all in the same plane.

Second law (Snell’s law):
The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given pair of media and for the given colour of light used. This constant is known as the refractive index of second medium w.r. t. the first medium.

Explanation:
If ‘i’ is the angle of incidence in the first medium and ‘r’ is the angle of refraction in the second medium, then by Snell’s law,

Where ₁n₂ is the refractive index of the second medium with respect to the first medium. If the first medium is air, then sini/sinr is known as absolute refractive index of the second medium.
ie, \(\frac{\sin i}{\sin r}=n\)
where ‘n’ is the refractive index of the second medium.

Some examples of refraction:
(a) Apparent depth:

When an object (in a denser medium) is viewed from a rarer medium, it seems to be raised towards the surface. This is called apparent depth.

(b) Twinkling of stars:
Twinkling of stars is due to the refraction of star light at different layers of the atmosphere. Due to this refraction the star at S appears at S¹. But the density of the layer continuously changes. So, the apparent position continuously changes. Thus the star appears to be twinkling.

(c) Apparent shift in the position of the sun at sunrise and sunset:

Sun is visible before sunrise and after sunset because of atmospheric refraction. The density of atmospheric air decreases as we go up. So the rays coming from the sun deviates towards the normal. So the sun at ‘S’ appears to come from ‘S¹’. Thus an observer on earth can see the sun before sunrise and after sunset.

Total Internal Reflection

When a ray of light passes from a denser to rarer medium, after refraction the ray bends away from the normal. If the angle of incidence increases, the angle of refraction increases. When the angle of refraction is 90°, the corresponding angle of incidence is called the critical angle.

If we increases the angle of incidence beyond the critical angle, the ray is totally reflected back to the same medium. This phenomenon is called total internal reflection.

Relation between critical angle and refractive index
Refractive index,

where ‘C’ is the critical angle.
A demonstration for total internal reflection
Demonstration – 1:

Take a soap solution in a beaker. Now direct the laser beam from one side of the beaker such that it strikes the upper surface of water obliquely. Adjust the direction of laser beam until the beam is totally reflected back to water.

Demonstration – 2:

Take a soap solution in a long test tube and shine the laser light from top, as shown in above figure. Adjust the direction of the laser beam such that it is totally internally reflected. This is similar to what happens in optical fibres.
Condition for total internal reflection:

1. Light should travel from denser medium to rarer medium.
2. Angle of incidence in the denser medium should be greater than the critical angle.

Relative critical angle:
Critical angle of a medium A with respect to a rarer medium B is represented as _BC_A. _BC_A is related to the refractive index _Bn_A as

Some Effects And Applications Of Total Internal Reflection
(a) Brilliance of diamond:
Refractive index of diamond is high (n = 2.42) and the critical angle is small (C = 24.41°). More over the faces of the diamond are cut in such a way that a ray of light entering the crystal undergoes multiple total reflections. This multiple reflected light come out through one or two faces. So these faces appear glittering.

(b) Mirage:
On hot summer days the layer of air in contact with the sand becomes hot and rare. The upper layers are comparatively cooler and denser. When light rays travel from denser to rarer, they undergo total internal reflection. Thus image of the distant object is seen inverted. This phenomenon is Known as mirage.

(c) Looming (superior mirage):
Due to the mist and fog in cold countries, distant ship cannot be seen clearly. But due to the total internal reflection, the image of the ship appears hanging in air. This illusion is known as looming.

(d)Total reflection prisms:

A right-angled prism is called a total reflecting prisms. Total reflecting prisms are based on the principle of total internal reflection. With the help of these prisms, the direction of the incident ray can be changed. The refractive index for glass is 1.5 and its critical angle is 42°. When a ray of light makes an angle of incident more than 42° (within the glass) the ray undergoes total internal reflection.

1. Optical fibres:

Optical fibres consist of a number of long fibres made of glass or quartz (n = 1.7). They are coated with a layer of a material of lower refractive index (1.5). When light incident on the optical fibre at angle greater than the critical angle, it undergoes total internal reflection. Due to this total internal reflection, a ray of light can travel through a twisted path.
Uses:

• Used as a light pipe in medical and optical diagnosis.
• It can be used for optical signal transmissions.
• Used to carry telephone, television and computer signals as pulses of light.
• Used for the transmission and reception of electrical signals which are converted into light signals.

Refraction At Spherical Surfaces And By Lenses
Spherical lenses:
There are two types of lenses

• convex lenses and
• concave lenses.

Principal axis:
A straight line passing through the two centers of curvature is called the principal axis of the lens.

Principal focus (F):
A narrow beam of parallel rays, parallel and close to the principal axis, after refraction, converges to a point on the principal axis in the case of a convex lens or appears to diverge from a point on the axis in the case of a concave lens. This fixed point is called the principal focus of the lens.

Focal length:
It is distance between the optic centre and the principal focus.

1. Refraction at a spherical surface:

Consider a convex surface XY, which separates two media having refractive indices n₁ and n₂. Let C be the centre of curvature and P be the pole. Let an object is placed at ‘O’, at a distance ‘u’ from the pole. I is the real image of the object at a distance V from the surface. OA is the incident ray at angle ‘i’ and Al is the refracted ray at an angle ‘r’. OP is the ray incident normally. So it passes without any deviation. From snell’s law,

r₁ = n₂ _____(1)
From the Δ OAC, exterior angle = sum of the interior opposite angles
i.e., i = α + θ ______(2)
Similarly, from ΔIAC,
a = α + β
r = α – β ______(3)
Substituting the values of eq(2) and eq(3)in eqn.(1) we get,
n₁(α + θ) = n₂(α – β)
n₁α + n₁θ = n₂α – n₂β
n₁θ + n₂β = n₂α – n₁α
n₁θ + n₂β = (n₂ – n₁)α _______(4)
From OAP, we can write,

From IAP, β = \(\frac{\mathrm{AP}}{\mathrm{PI}}\), From CAP, α = \(\frac{\mathrm{AP}}{\mathrm{PC}}\)
Substituting θ, β and α in equation (4) we get,

According to New Cartesian sign convection, we can write,
OP = -u, PI = +v and PC = R
Substituting these values, we get

Case -1: If the first medium is air, n₁ = 1, and n₂ = n,

2. Refraction by a lens:
Lens Maker’s Formula (for a thin lens):
Consider a thin lens of refractive index n₂ formed by the spherical surfaces ABC and ADC. Let the lens is kept in a medium of refractive index n₁ Let an object ‘O’ is placed in the medium of refractive index n₁ Hence the incident ray OM is in the medium of refractive index n₁ and the refracted ray MN is in the medium of refractive index n₂.

The spherical surface ABC (radius of curvature R₁) forms the image at I₁. Let ‘u’ be the object distance and ‘v₁‘ be the image distance.
Then we can write,

This image I₁ will act as the virtual object for the surface ADC and forms the image at v.
Then we can write,

Adding eq (1) and eq (2) we get

Dividing throughout by n1, we get

if the lens is kept in air, \(\frac{\mathrm{n}_{2}}{\mathrm{n}_{1}}\) = n
So the above equation can be written as,

From the definition of the lens, we can take, when u = 8, f = v
Substituting these values in the eq (3), we get


For convex lens,
f = +ve, R₁ = +ve, R₂ = – ve

For concave lens,
f = -ve, R₁ = -ve, R₂ = +ve

Lens formula

Linear magnification: If h_o is the height of the object and h_i is the height of the image, then linear magnification

3. Power of a lens:
Power of a lens is the reciprocal of focal length expressed in meter.

Unit of power is dioptre (D).

4. Combination of thin lenses in contact:

Consider two thin convex lenses of focal lengths f₁ and f₂ kept in contact. Let O be an object kept at a distance ‘u’ from the first lens L₁, I₁ is the image formed by the first lens at a distance v₁.
Then from the lens formula, we can write,

This image will act as the virtual object for the second lens and the final image is formed at I (at a distance v). Then

If the two lenses are replaced by a single lens of focal length ‘F’ the image is formed at V. Then we can write,

where P is the power of the combination, P₁ and P₂ are the powers of the individual lenses.

Magnification (combination of lenses):
If m₁, m₂, m₃,…….. are the magnification produced by each lens,
then the net magnification,
m = m₁. m₂. m₃……….
Relation connecting m, u and f:

Relation connecting m,v and f:

Refraction Through A Prism

ABC is a section of a prism. AB and AC are the refracting faces, BC is the base of the prism, ∠A is the angle of prism.
Aray PQ incidents on the face AB at an angle i₁. QR is the refracted ray inside the prism, which makes two angles r₁ and r₂ (inside the prism). RS is the emergent ray at angle i₂.
The angle between the emergent ray and incident ray is the deviation ‘d’.
In the quadrilateral AQMR,
∠Q + ∠R = 180°
[since and N₁M are normal] ie,
∠A + ∠M = 180° ____(1)
In the Δ QMR.
∴ r₁ + r₂ + ∠M = 180° _____(2)
Comparing eq (1) and eq (2)
r₁ + r₂ = ∠A ______(3)
From the Δ QRT,
(i₁ – r₁) + (i₂ – r₂) = d
[since exterior angle equal sum of the opposite interior angles]
(i₁ + i₂) – (r₁ + r₂) = d
but, r₁ + r₂ =A
∴ (i₁ + i₂ ) – A = d
(i₁ + i₂) = d + A _____(4)
It is found that for a particular angle of incidence, the deviation is found to be minimum value ‘D’.
At the minimum deviation position,
i₁ = i₂ = i, r₁ = r₂ = r and d = D
Hence eq (3) can be written as,
r + r= A
or r = \(\frac{A}{2}\) ______(5)
Similarly eq (4) can be written as,
i + i = A + D
i = \(\frac{A+D}{2}\) _____(6)
Let n be the refractive index of the prism, then we can write,
n = \(\frac{\sin i}{\sin r}\) ______(7)
Substituting eq (5) and eq (6) in eq (7),

i – d curve:

It is found that when the angle of incidence increases deviation (d) decreases and reaches a minimum value and then increases. This minimum value of the angle of deviation is called the angle of minimum deviation.

Dispersion By A Prism
Dispersion: The splitting of the white light into its component colours is called dispersion.

The pattern of colour components of light is called the spectrum of light.

Reason for dispersion:
The refractive index is different for different colours. Refractive index for violet is higher than red. This variation of refractive index of medium with the wavelength causes dispersion.

Some Natural Phenomena Due To Sunlight
1. The rainbow:
The rainbow is an example of the dispersion of sunlight by the water drops in the atmosphere. The conditions for observing a rainbow are that the sun should be shining in one part of the sky while it is raining in the opposite part of the sky.
There are two types rainbow

• Primary rainbow
• secondary rainbow.

(i) Primary rainbow:

In a primary rainbow, after refraction at the surface of water droplet, the ray suffers one internal reflection and finally comes out of the drop by forming an inverted spectrum. The maximum deviated light is red (42°) and the least deviated light is violet (40°).

(ii) Secondary rainbow:

secondary rainbow, after refraction at the surface of water droplet, the ray suffers two total internal reflection and finally comes out of the droplet by forming a spectrum. The most deviated light in this spectrum is violet (53°) and the least deviated light is red (50°).

2. Scattering of light:
When sunlight travels through the earth’s atmosphere, it changes its direction by atmospheric particles. This is called scattering. Light of shorter wavelength is scattered much more than light of longer wavelength. Scattering is possible only when size of the particles is comparable to the wavelength of incident light.

Rayleigh’s scattering law:
The intensity of the scattered light from a molecule is inversely proportional to the 4^th power of the wavelength.
ie, \(I \alpha \frac{1}{\lambda^{4}}\)
I – Intensity of Scattering

Blue colour of sky:
According to Rayleigh scattering, scattering is inversely proportional to the fourth power of its wavelength. Hence shorterwavelength is scattered much more than longer wavelength. Thus blue colour is more scattered than the other colours. So sky appears blue.

Whiteness of clouds:
Clouds contain large partides (dust, H₂O), which scatter all colours almost equally. Hence clouds appear white.

Colours of the sunset (or sunrise):
At sunrise and sunset light has to travel a longer distance before reaching the earth. During this time, smaller wavelengths are scattered away. The remaining colours is red. Hence sky appears red in colour.

Optical Instruments
Mirrors, lenses and prisms, periscope, Kaleidoscope, Binoculars, telescopes, microscopes are some examples of optical devices Our eye is one of the most important optical device.

1. The eye:

Human eye consists of an eyeball of size 2.5cm in diameter. The very thin skin in front of the eye is known as cornea. Behind cornea, the empty space is known as aqueous humor. The small wall behind cornea is known as iris. In this iris a small circular opening is there, which is known as pupil. Iris can adjust its tension to vary the size of the pupil.

Behind the iris a muscular membrane is there which is known as ciliary muscle. The focal length of the crystalline lens can be adjusted to see the object any separation by adjusting the tension of ciliary muscles. The backwall of eye is known as retina.

It consists of light sensitive cells known as rods and cones. The rods are sensitive to intensity and cones are sensitive to colour. The signals from retina are transferred to the brain by optic nerves.

The brightest point in the retina is known as yellow spot and the lowest point in the eye (retina) is known as blind spot. The space between the lens and retina is filled by a liquid which is known as vitreous humor.

Defects of Vision
a. Myopia or shortsightedness:

A person suffering from myopia can see only nearby objects but cannot see objects beyond a certain distance clearly. This defect occurs due to

1. Elongation of eyeball
2. Short focal length of eye lens

It can be corrected by using a concave lens of suitable focal length.

b. Hypermetropia or Far sightedness:

A person suffering from this defect can see only distant object clearly but cannot see nearby objects clearly. This defect occurs due to

1. Decrease in the size of the eyeball.
2. Increase in focal length of the eyeball.

This defect can be corrected by using a converging lens (convex).

c. Astigmatism:

A person suffering from astigmatism cannot focus objects in front of the eye clearly. It can be corrected by using a cylindrical lens of suitable focal length.

d. Presbyopia:
It is the farsightedness occurring due to theawakening of the ciliary muscles. It can be corrected by using a lens of bifocal length.

1. The microscope:
Simple microscope: A simple microscope is a converging lens of small focal length.

Working: The object to be magnified is placed very close to the lens and the eye is positioned close to the lens on the other side. Depending upon the position of object, the position of image is changed.

Case 1:
If the object is placed, one focal length away or less, we get an erect, magnified and virtual image at a distance so that it can be viewed comfortably ie. at 25cm or more. (This 25cm is denoted by the symbol D).

Case 2:
If the object is placed at a distance f (focal length of lens), we get the image at infinity.

The mathematical expression of magnification:
Image at D:
If the image is formed at ‘D’, we can take u = -D. Hence the lens formula can be written as

The image is formed at D, ie. v = -D

This equation is used to find magnification of simple microscope when image at D (D ≈ 25cm).

Image at infinity:

If the object is placed at f, the image forms at infinity. In this case, magnification,

Suppose the object has a height h, the angle subtended is
tanθ₀\(=\frac{h}{D}\), θ₀\(=\frac{h}{D}\)______(2)
where ‘D’ is the comfortable distance of object from the eye (least distinct vision).

When the final image is formed at infinity,
θ_i = \(\frac{h^{1}}{v}\) ______(3)
When h₁ is the height of image and v is the image distance

This equation is used to find magnification of simple microscope when image at infinity.

2. Compound microscope:
Apparatus: A compound microscope consists of two convex lenses, one is called the objective and the other is called eye piece.

The convex lens near to the object is called objective. The lens near to the eye is called eye piece. The two lenses are fixed at the ends of two co-axial tubes. The distance between the tubes can be adjusted.

Working:
The object is placed in between F and 2F of objective lens. The objective lens forms real inverted and magnified image (I₁M₁) on the other side of the lens.

This image will act as object or eyepiece. Thus an enlarged, virtual, and inverted image is formed, (this image can be adjusted to be at the least distance of distinct vision, D).

Magnification: The magnification produced by the compound microscope

Where m₀ & m_e are the magnifying power of objective lens and eyepiece lens.

Eyepiece acts as a simple microscope.
Therefore m_e = 1 + \(\frac{D}{f_{e}}\) _____(2)
m₀ = \(\frac{v_{0}}{u_{0}}\) ______(3)
We know magnification of objective lens
Where v₀ and u₀ are the distance of the image and object from the objective lens.
Substituting (2) and (3) in (1), we get

for compound microscope, u_o » f_o (because the object of is placed very close to the principal focus of the objective) and v_o ≈ L, length of microscope (because the first image is formed very close to the eye piece).

where L is the length of microscope, f₀ is the focal length of objective lens.
Case 1: If the final image is formed at infinity, magnification of eye piece D
m \(=\frac{D}{f_{e}}\)
∴ Total magnification of compound microscope

(3) Telescope: Astronomical telescope is used to observe heavenly bodies.
There are two types of telescopes

1. Refracting and
2. Reflecting Telescope.

(1) Refracting Telescope:
Constructional details

It consists of two convex lenses, one is called objective and other is called eyepiece. These two lenses are fitted at the ends of two coaxial tubes. The distance between the two lenses can be varied.

Working:
The objective lens forms the image (IM) of a distant object at its focus. This image (formed by objective) is adjusted to be focus of the eyepiece.

Magnification:
The magnifying power of a telescope is the ratio of the angle subtended by the image at the eye to the angle subtended by the object at the objective.

(For small values tan α ≈ α)

But IC = f_o (the focal length objective lens) and IC¹ = f_e(the focal length eyepiece lens.)
∴ m = \(\frac{f_{0}}{f_{e}}\)
In this case the length of the telescope tube is (f₀ + f_e).

Case 1: When the image formed by the objective is within the focal length of the eyepiece, Then the final image is formed at the least distant of distinct vision. In this case, magnifying power.

(2) Reflecting Telescope:
Newtonian types reflecting Telescope:
The Newtonian reflector consists of a parabolic mirror made of an alloy of copper and tin. It is fixed atone end of a metal tube.

The parallel rays from a distant stars incident on the mirror M₁. After reflection from the mirror, the ray incident on a plane mirror M₂.

The reflected ray from M₂ enter into eye piece E. The eyepiece forms a magnified, virtual and erect image. Magnifying power of Newton Telescope
m = \(\frac{f_{0}}{f_{e}}\) or m = \(\frac{R}{2 f_{\theta_{g}}}\)
where
f_o — is the focal length of concave mirror
f₂ — is the focal length of eyepiece.
R – Radius of curvature of concave reflector.

We hope the Plus Two Physics Notes Chapter 9 Ray Optics and Optical Instruments help you. If you have any query regarding Plus Two Physics Notes Chapter 9 Ray Optics and Optical Instruments, drop a comment below and we will get back to you at the earliest.

Plus Two Physics Notes Chapter 8 Electromagnetic Waves is part of Plus Two Physics Notes. Here we have given Plus Two Physics Notes Chapter 8 Electromagnetic Waves.

Kerala Plus Two Physics Notes Chapter 8 Electromagnetic Waves

Introduction
In this chapter, we shall study the basic concepts of electromagnetic waves.

Displacement Current
Amperes circuital law in ac circuit: Consider a capacitor connected to a AC source using conducting wires. AC current can flow through a capacitor. Hence magnetic field is produced around the conducting wire. This magnetic field can be found using amperes circuital law.

Magnetic field at P
Method – 1 (To find magnetic field at P)

Consider a point P, which lies outside and very close to a capacitor as shown in the figure. We can find magnetic field at P using amperes circuital law. In order to find an magnetic field at P, consider a open surface (amperien loop having pot like surface) with a boundary of circle of radius r.
Applying amperes circuital law we get
\(\oint\)B.dI = µ₀i
Where ‘i’ is the current passing through the surface. (This surface lies outside to capacitor)
Integrating we get B.2πr = µ₀i
B = \(\frac{\mu_{0} i}{2 \pi r}\) _____(1)

Method – 2 (To find magnetic field at P)

Consider a open surface (amperien loop having pot like surface) extended to interior of capacitor with a boundary of circle of radius r.
Applying amperes circuital law we get
\(\oint\)B.dI = µ₀0
(since the current passing through the closed surface is zero, surface lies in between the plates)
ie. B = 0 ______(2)

Discussion of method 1 and method 2: Amperian circuital law is independent of size and shape of pot like surface. Hence we expect same value of B in eq(1)and eq(2). But we got different values at the same point P. Hence we can understand that there is a mistake in the amperes circuital law in AC circuits.

Maxwells correction in amperes circuital law:
To solve the above mistake, Maxwell introduced a term in the amperes circuital law. The modified amperes circuital law can be written as
\(\oint\)B.dI = µ₀(i_c + i_d)
Where i_d is called displacement current. Its value is

The above-modified amperes circuital is known as Ampere- Maxwell law. This law is applicable for both AC and DC circuits.

Question 1.
Show that conduction current i_c is equal to displacement current i_d
Answer:
The flux passing through the surface in between plates (see figure 2)

Capacitor is connected to ac voltage. Hence the charge on the plate also changes with time. Hence the flux passing through the pot shape surface changes with time.
ie. the flux in between capacitor changes.
The change influx,

This means that the conduction current passing through the conduction wire is converted into displacement current, when it passes in between plates of capacitor.
1. The total current i is the sum of the conduction current and the displacement current
So we have

2. Outside the capacitor plates, we have only conduction current and no displacement current inside the capacitor there is no conduction current and there is only displacement current.

Electromagnetic Waves
1. Sources of Electromagnetic waves:
Question 2.
How are electromagnetic waves produced?
Answer:
Consider a charge oscillating with some frequency (An oscillating charge is an example of accelerating charge). This oscillation produces an oscillating electric and magnetic field in space. The oscillating electric and magnetic fields (EM Wave) propagates through the space. The experimental production of electromagnetic wave was done by Hertz’s experiment in 1887.

2. Nature of electromagnetic waves:
Characteristics of Electromagnetic waves:
(i) Electromagnetic waves propagate in the form of mutually perpendicular magnetic and electric
fields. The direction of propagation of wave is perpendicular to both magnetic and electric field vector.

(ii) Velocity of electromagnetic waves in free space is,

The speed of electromagnetic wave in a material medium is given by

(iii) The ratio of magnitudes of electric and magnetic field vectors in free space is constant

E and B are in same phase

(iv) No medium is required for propagation of transverse wave.

(v) Electromagnetic waves show properties of reflection, refraction, interference, diffraction and polarization.

(vi) Electromagnetic waves have capability to carry energy from one place to another.

Mathematical Expression:

Consider a plane electromagnetic wave travelling along the Z direction. The electric and magnetic fields are perpendicular to the direction of wave motion.
The electric field vector along the Y direction.
E_x = E₀sin(kz – ωt)
and B_Y = B₀sin(kz – ωt)
where E₀ is the amplitude of electric field vector, B₀ is the amplitude of magnetic field vector, ω is the angular frequency and k is related to the wavelength λ of the wave,
k = 2π/λ.

Electromagnetic Spectrum
Electromagnetic waves include visible light waves, X-rays, gamma rays, radio waves, microwaves, ultraviolet and infrared waves. The classification is based roughly on how the waves are produced or detected.

1. Radio waves:
Radio waves are produced by the accelerated motion of charges in conducting wires. They are used in radio and television communication systems. They are generally in the frequency range from 500 kHz to about 1000 MHz.

2. Microwaves:
Microwaves (short-wavelength radio waves), with frequencies in the gigahertz (GHz) range, are produced by special vacuum tubes (called klystrons, magnetrons, and Gunn diodes). Due to their short wavelengths, they are suitable for the radar systems used in aircraft navigation. Microwave ovens are domestic application of these waves.

3. Infrared waves:
Infrared waves are produced by hot bodies and molecules. Infrared waves are sometimes referred to as heatwaves. Infrared lamps are used in physical therapy.

Infrared rays are widely used in the remote switches of household electronic systems such as TV, video recorders etc. Infrared radiation also plays an important role in maintaining the earth’s warmth or average temperature through the greenhouse effect.

4. Visible rays:
It is the part of the spectrum that is detected by the human eye. It starts from 4 × 10¹⁴ Hz to 7 × 10¹⁴ Hz (ora wavelength range of about 700 – 400 nm).

5. Ultraviolet rays (UV):
It covers wavelengths ranging from about 4 × 10^-7m to 6 × 10^-10m (0.6 nm to 400 nm)). UV radiation is produced by special lamps and very hot bodies. The sun is an important source of ultraviolet light.

UV light in large quantities has harmful effects on humans. Exposure to UV radiation induces the production of more melanin, causing tanning of the skin. UV radiation is absorbed by ordinary glass. Hence, one cannot get tanned or sunburn through glass windows.

Due to its shorter wavelengths, UV radiations can be focussed into very narrow beams for high precision applications such as eye surgery. UV lamps are used to kill germs in water purifiers.

6. X-rays:
It covers wavelengths from about 10^-8m to 10^-13m (4nm – 10nm). One common way to generate X-rays is to bombard a metal target by high energy electrons. X-rays are used as a diagnostic tool in medicine and as a treatment for certain forms of cancer.

7. Gamma rays:
They lie in the upper-frequency range of the electromagnetic spectrum and have wavelengths of^rom about 10^-10m to less than 10^-14m. This high-frequency radiation is produced in nuclear reactions and also emitted by radioactive nuclei. They are used in medicine to destroy cancer cells.

Different Types Of Electromagnetic Waves:

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Plus Two Physics Notes Chapter 7 Alternating Current is part of Plus Two Physics Notes. Here we have given Plus Two Physics Notes Chapter 7 Alternating Current.

Kerala Plus Two Physics Notes Chapter 7 Alternating Current

Alternating Current
AC current is commonly used in homes and offices. The main reason for preferring ac voltage over dc voltage is that ac voltages can be easily converted from one voltage to the other and can be transmitted over long distances. In this chapterwe will deal the properties of ac and its flowthrough different devices (inductor, capacitor, etc).

Ac Voltage Applied To A Resistor

Consider a circuit containing a resistance ‘R’ connected to an alternating voltage.
Let the applied voltage be
V = V_o sinωt ______(1)
According to Ohm’s law, the current at any instant can be written as
I = \(\frac{V_{0} \sin \omega t}{R}\)
Where I₀ = V_o/R is the peak value of current. Comparing eq(1) and eq(2), we can understand that the current and voltage are in same phase.
Graphical variation of current and voltage:

R.M.S value (or Virtual value, effective value) of current and voltage:
The mean value of emf and current for one cycle is zero. Hence to measure ac, the root mean square (rms) values are considered.

The r.m.s value or virtual value of an AC is the square root of the mean of the squares of the instantaneous value of current taken over a complete cycle.
I_rms = \(\frac{I_{0}}{\sqrt{2}}\) and V_rms = \(\frac{V_{0}}{\sqrt{2}}\)
where I₀ – maximum current, V₀ – maximum voltage, (r.m.s.- root mean square).

Power dissipated in the resistor:
The average power consumed in one complete cycle,

Substituting current and voltage, We get

Representation Of Ac Current And Voltage By Rotating Vectors – Phasors
To represent the phase relation between current and voltage, phasors are used. Aphasoris a vector which rotates about the origin with an angular speed ω. The vertical components of phasors of V and I represent instantaneous value of V and I at a time t (see figure). The length of phasors give maximum amplitudes of V and I.

Phasor diagram of v and i for the circuit containing resistor only

Figure (a) represent the voltage and current phasors and their relationship at time t₁. Fig (b) shows the graphical variation of V and I.

Ac Voltage Applied To An Inductor

Consider a circuit containing an inductor of inductance ‘L’ connected to an alternating voltage.
Let the applied voltage be
V = V_o sinωt _____(1)
Due to the flow of alternating current through coil, an emf, \(\frac{d I}{d t}\) is produced in the coil. This induced emf is equal and opposite to the applied emf (in the case of ideal inductor)

Integrating, we get

Where I_o = \(\frac{V_{0}}{L \omega}\),
The term Lω is called inductive reactance. Comparing eq(1) and eq(2), we can understand that, the current lags behind the voltage by an angle 90°.
Graphical variation of current and voltage:

Phasor diagram:

Inductive reactance X_L:
The resistance offered by an inductor to a.c. flow is called inductive reactance.
Inductive reactance

Power Consumed by an Inductor Carrying AC:
The instantaneous value of voltage and current in a pure inductor is
V = V_o sinωt
I = I_o cosωt
The average power consumed per cycle.

The above expression indicates that the average power or net energy consumed by an inductor carrying ac for a full cycle is zero.

Ac Voltage Applied To A Capacitor

Consider a circuit containing a capacitor of capacitance ‘C’ connected to alternating voltage.
Let the applied voltage be V = V_o sinωt _____(1)
The instantaneous current through capacitor

Substituting eq.(1) in eq.(2), we get

\(\frac{1}{\mathrm{C} \omega}\) is called capacitative reactance
Comparing eq(1) and eq(3), we can understand that, the current leads the voltage by an angle 90°
Graphical variation of current and voltage:

Phaser diagram:

Capacitative Reactance X_c:
The resistance offered by a capacitor to ac flow is called Capacitative reactance
Capacitative reactance

1. Power consumed by a capacitor carrying current:
The instantaneous value of voltage and current in a pure inductor is
V = V_o sinωt
I = I_o cosωt
The average power consumed per cycle.

The above expression indicates that the average power or net energy consumed by a capacitor carrying ac for a full cycle is zero.

Ac Voltage Applied To A Series Lcr Circuit

Consider a circuit containing an inductance L, resistance R and capacitance C connected in series. An alternating voltage V = V_o sinωt is applied to the circuit.
Phasor Diagram:

Let V_R be the voltage across R. This voltage is represented by a vector OA (since I and V_R are in same direction). Let V_L be the voltage across L. This voltage is represented by a vector OB (since the voltage V_L leads the current by angle 90°).

Similarly, let V_c be the voltage across C. This voltage is represented by a vector OC (since the voltage V_c lags the current by angle 90°).

The phase difference between V_L and V_c is Φ(ie. they are in opposite directions). So the magnitude of net voltage across the reactance is (V_L – V_c). This is represented by a vector OD in phasor diagram.

The final voltage in the circuit is the vector sum of V_R and (V_L – V_c). The final voltage is represented by diagonal OE.

1. Impedances of LCR circuit:
From the right angled triangle OAE,
Final voltage, V = \(\sqrt{\mathrm{V}_{\mathrm{R}}^{2}+\left(\mathrm{V}_{\mathrm{L}}-\mathrm{V}_{\mathrm{c}}\right)^{2}}\)

Where Z is called impedance of LCR circuit

Phase Difference: Let Φ be the phase difference between final voltage V and current I
From fig (2), we can write

Expression for current:
The eq(2) shows that there is a phase difference between current and voltage. The instantaneous current lags the voltage by an angle (Φ).
If V = V_o sinωt is the applied voltage, the current at any instant can be written as
I = I_o sin(ωt – Φ) _____(3)
Where I_o is the peak value of current. It’s value can be written as

2. Analytical solution:
If we apply V = V_msinωt to an LCR circuit, we can write
V_L + V_R + V_C = V_m sinωt

Substituting these values in eq.(1), we get

The above equation (2) is like the equation for a forced, damped oscillator. Hence we can take the solution of above equation as
q = q_m sin(ωt + θ)

Substituting these values in eq.(2) we get

Multiplying and dividing by Z = \(\sqrt{R^{2}+\left(X_{0}-X_{L}\right)^{2}}\), we have

Substituting these values in eq.(4), we get
q_mωz[cosΦcos(ωt + θ) + sinΦsin(ωt + θ)] = V_msinωt
q_mωz cos(ωt + θ – Φ) = V_m sinωt ______(5)
(∴ cos(A – B) = cosA cosB + sinA sinB)
Comparing the two sides of the eq.(5) we get
V_m = q_mωz = i_mz
where i_m = q_mω
and cos(ωt + θ – Φ) = sinωt
sin (ωt + θ – Φ + π/2) = sinωt (∵ sin(θ + π/2) = cosθ)
ωt + θ – Φ + π/2 = ωt
(θ – Φ) = -π/2
Therefore, the current in the circuit is

Thus, the analytical solution for the amplitude and phase of the current in the circuit agrees with that obtained by the technique of phasors.

3. Resonance:
When ωL = \(\frac{1}{\omega C}\), the impedance of the LCR circuit becomes minimum. Hence current becomes maximum. This phenomena is called resonance.

The frequency of the applied signal at which the impedance of LCR circuit is minimum and current becomes maximum is called resonance frequency.
Expression for resonance frequency
Resonance occurs at ωL = \(\frac{1}{\omega C}\)

Impedance at resonance: Resonance occurs at ωL = \(\frac{1}{\omega C}\). Substituting this condition in eq(1), in section 7.6, we get
Impedance, Z = R
Current at resonance:
substituting ωL = \(\frac{1}{\omega C}\) in eq(2) in section 7.6 we get,
TanΦ = 0, or Φ = 0
Substituting this value in eq(3) in section 7.6, we get
current I= I_o sin ωt
Where I_o = V_o/R
Graphical variation of current with ω in LCR circuit:

Variation of current through LCR circuit with angular frequency co for two cases (1) R= 100Ω and (2) R=200Ω, is shown in the graph
Note: It is important to note that resonance phenomenon is exhibited by a circuit only if both L and C are present in the circuit. Only then the voltages across L and C cancel each other (both being out of phase).

The current amplitude ( V_m/R) is the total source voltage appearing across R. This means that we cannot have resonance in a RL or RC circuit.

4. Sharpness of resonance:

The figure shows the variation of current i with © in a LCR circuit.
Bandwidth: At ω₀, the current in the LCR circuit is maximum. Suppose we choose a value of ω for which the current amplitude is \(\frac{1}{\sqrt{2}}\) times its maximum value.

We can see that there are two values of ω(ω₁ and ω₂) and forwhich current is \(\frac{i_{m}}{\sqrt{2}}\).

The difference ω₂ – ω₁ is called bandwidth.
If we take ω₁ = ω₀ – ∆ω and ω₂ = ω₀ + ∆ω
We get bandwidth, ω₂ – ω₁ = 2∆ω.

Expression for bandwidth and sharpness of resonance:
We know that the current in the LCR circuit

We know that the current in the LCR circuit becomes \(\frac{i_{m}}{\sqrt{2}}\) at ω₂ = ω₀ + ∆ω. Substituting this m in eq(1), we get

But ω₂ = ω₀ + ∆ω, substituting this above equation.


Sharpness of resonance: The quantity \(\left(\frac{\omega_{0}}{2 \Delta \omega}\right)\) is
called sharpness of resonance.
From eq.(4),weget

When bandwidth increases, the sharpness of resonance decreases, ie. the tuning of the circuit will not be good.

Quality Factor (Q): The ratio \(\frac{\omega_{0} L}{R}\) is called the quality factor. When R is low or L is large, the quality factor becomes large. Lange quality factor means that the circuit is more selective.

Power in AC circuit: The power factor
Power in AC circuit with LC and R: In ac circuit the Voltage vary continuously.
∴ The average power in the circuit for one full cycle of period,


(since sin 2A = 2sinA CosA)
The mean value of sin²ωt over a complete cycle is 1/2 and the mean value of sin2ωt over a complete cycle is zero.

True power = Apparent power × power factor
The term P_av called true power. V_rms × I_rms is called the apparent power and cosΦ is called power factor.
powerfactor = \(\frac{\text { True power }}{\text { apparent power }}\)
Power factor is defined as the ratio of true power to apparent power.

Case – 1 (In purely resistive circuit)
In this case, current and voltage are in same phase. Hence Φ = 0
∴ P_av = V_rms I_rmsCosO
True power, P_av = V_rms I_rms

Case – 2 (In a purely inductive and purely capacitative circuit (no resistance)). In this case, the angle between voltage and current is 90°.
∴ P_av = V_rms I_rmsCos 90
True power, P_av = 0
Which means that the power consumed by the circuit is zero. The current in such a circuit (purely inductive and purely capacitive) doesn’t do any work. A current that does not do any work is called wattles or idle current.

Lc Oscillations

A capacitor can store electrical energy. An inductor can store magnetic energy. When a charged capacitor is connected to an inductor, the electrical energy( of capacitor) transfers to magnetic energy (of inductor) and vise versa. Thus energy oscillates back and forth between capacitor and inductor. This is called L. C. Oscillations.
Expression for frequency:
Applying Kirchoff’s second rule, we get

Transformers
Principle: It works on the principle of mutual induction.
Construction:

A transformer consists of two insulated coils wound over a core. The coil, to which energy is given is called primary and that from which energy is taken is called secondary.

Working and mathematical expression :
Let V₁ N₁ be the voltage and number of turns in the primary. Similarly, let V₂, N₂ be voltage and number of turns in the secondary.

When AC is passed, a change in magnetic flux is produced in the primary. This magnetic flux passes through secondary coil.

If Φ₁ and Φ₂ are the magnetic flux of primary and secondary, we can write Φ₁ α N₁ and Φ₂ α N₂.
Dividing Φ₁ and Φ₂
\(\frac{\phi_{1}}{\phi_{2}}=\frac{N_{1}}{N_{2}}\)
[since Φ is proportional to number of turns] or \(\phi_{1}=\frac{\mathrm{N}_{1}}{\mathrm{N}_{2}} \phi_{2}\)
Taking differentiation on both sides we get

Step up Transformer:
If the output voltage is greater than input voltage, the transformer is called step up transformer. In a step up transformer N₂ > N₁ and V₂ > V₁.

Step down transformer:
If the output voltage is less than the input voltage, then the transformer is called step down transformer. In a step down transformer N₂ < N₁ and V₂ < V₁.

Efficiency of a transformer:
The efficiency of a transformer is defined as the ratio of output power to input power.

For an ideal transformer, efficiency = 1
i.e, V₁I₁ = V₂I₂

1. Power losses in a transformer
(i) Joule loss or Copper loss:
When current passes through a coil heat is produced. This energy loss is called Joule loss. It can be minimized by using thick wires.

(ii) Eddy current loss: This can be minimized by using laminated cores. Laminated core increases the resistance of the coil. Thus eddy current decreases.

(iii) Hysteresis loss: When the iron core undergoes cycles of magnetization, energy is lost. This loss is called hysteresis loss. This is minimized by using soft iron core.

(iv) Magnetic flux loss:
The total flux linked with the coil may not pass through secondary coil. This loss is called magnetic flux loss. This loss can be minimized by closely winding the wires.

We hope the Plus Two Physics Notes Chapter 7 Alternating Current help you. If you have any query regarding Plus Two Physics Notes Chapter 7 Alternating Current, drop a comment below and we will get back to you at the earliest.

Plus Two Physics Notes Chapter 6 Electromagnetic Induction is part of Plus Two Physics Notes. Here we have given Plus Two Physics Notes Chapter 6 Electromagnetic Induction.

Kerala Plus Two Physics Notes Chapter 6 Electromagnetic Induction

Introduction
In this chapter we are going to discuss the laws governing electromagnetic induction; how energy can be stored in a coil, the generation of ac, the relation between voltage and current in various circuit components, and finally the working of the transformer.

The Experiments Of Faraday And Henry
Faraday and Henry conducted a series of experiments to develop principles of electromagnetic induction.
Experiment – 1

Connect a coil to a galvanometer G as shown in the figure. When the north pole of a bar magnet is pushed towards the coil, galvanometer shows deflection. The deflection indicates that a current is produced in the coil.

The galvanometer does not show any deflection when the magnet is held stationary. When the magnet is pulled away from the coil, the galvanometer shows deflection in the opposite direction.

Conclusion of experiment 1:
The relative motion between magnet and coil produces an electric current in the first coil.
Experiment 2

Connect a coil C₁ to a galvanometer G. Take another coil C₂ and connect it with a battery. A steady current in the coil produces a steady magnetic field. When the coil C₂ is moved towards the coil C₁, the galvanometer shows a deflection. This deflection indicates that the electric current is induced in the coil G.

When the coil C₂ is moved away, the galvanometer shows a deflection in the opposite direction. When the coil C₂ is kept fixed, no deflection is produced in the coil C₁.
Conclusion of Experiment – 2:
The relative motion between two coils induces an electric current in the first coil.
Experiment – 3

In this experiment coil C₁ is connected to galvanometer G. The second coil C₂ is connected to a battery through a key K.

When the key K is pressed, the galvanometer shows a deflection. If the key is held pressed continuously, there is no deflection in the galvanometer. When the key is released, a momentary deflection is observed again, (but in opposite direction).

Conclusion of experiment – 3:
The change in current in second coil induces a current in the first coil.

Magnetic Flux

Magnetic flux through a plane of area A placed in uniform magnetic field B can be written as

Φ = BAcosθ

Faraday’S Law Of Induction
Faraday’s law of electromagnetic induction states that the magnitude of the induced emf in a circuit is equal to the time rate of change of magnetic flux through the circuit.
Mathematically, the induced emf is given by

If the coil contain N turns, the total induced emf is given by,

Lenz’S Law And Conservation Of Energy
Lenz’s law:
Lenz’s law states that the direction emf (or current) is such that it opposes the change in magnetic flux which produces it,
Mathematically the Lenz’s law can be written as

The negative sign represents the effect of Lenz’s law. The magnitude of the induced emf is given by Faraday’s law. But Lenz’s law gives the direction induced emf.
Lenz’s law is in accordance with the law of conservation of energy.

When the north pole of the magnet is moved towards the coil, the side of the coil facing north pole becomes north as shown in above figure. (Current is produced in the coil and flows in an anticlockwise direction).

So work has to be done to move a magnet against this repulsion. This work is converted into electrical energy.

When the north pole of the magnet is moved away from the coil, the end of the coil facing the north pole acquires south polarity. So work has to be done to overcome the attraction. This work is converted into electrical energy. This electrical energy is dissipated as heat produced by the induced current.

Motional Electromotive Force

Consider a rectangular frame MSRN in which the conductor PQ is free to move as shown in figure. The straight conductor PQ is moved towards the left with a constant velocity v perpendicular to a uniform magnetic field B. PQRS forms a closed circuit enclosing an area that change as PQ moves. Let the length RQ = x and RS = I.
The magnetic flux Φ linked with loop PQRS will be BIx.
Since x is changing with time the rate of change of flux Φ will induce an e.m.f. given by

Energy Consideration: A Quantitative Study
Let ‘r’ be the resistance of arm PQ. Consider the resistance of arm QR, RS, and SP as zero. When the arm is moved,
The current produced in the loop,

This current flows through the arm PQ. The arm PQ is placed in a magnetic field. Hence force acting on the arm,

Substituting eq. (1) in eq. (2)

If this arm is pulling with a constant velocity v, the power required for motion P = Fv

The external agent that does this work is mechanical. Where does this mechanical energy go?
This energy is dissipated as heat. The power dissipated by Joule law,

From eq. (3) and (4), we can understand that the workdone to pull the conductor is converted into heat energy in conductor.
Relation between induced charge and magnetic flux:
We know

so the above equation can be written as

We also know magnitude of induced emf

Comparing (1) and (2), we get

Eddy Currents
Whenever the magnetic flux linked with a metal block changes, induced currents are produced. The induced currents flow in closed paths. Such currents are called eddy currents.
Experiment to Demonstrate Eddy Currents
Experiment -1

Allow a rectangular metal sheet to oscillate in between the pole pieces of strong magnet as shown in figure. When the plate oscillates, the magnetic flux associated with the plate changes. This will induce eddy currents in the plate. Due to this eddy current the rectangular metal sheet comes to rest quickly.
Experiment – 2

Make rectangular slots on the copper plate. These slots will reduce area of plate. Allow this copper plate to oscillate in between magnets. Due to this decrease in area, the eddy current is also decreased. Hence the plate swings more freely.

Some important applications of Eddy Currents:
1. Magnetic braking in trains:
Strong electromagnets are situated above the rails. When the electromagnets are activated, eddy currents induced in the rails. This eddy current will oppose the motion of the train.

2. Electromagnetic damping:
Certain galvanometers have a core of metallic material. When the coil oscillates, the eddy currents are generated in the core. This eddy current opposes the motion and brings the coil to rest quickly.

3. Induction furnace:
Induction furnace can be used to melt metals. A high frequency alternating current is passed through a coil. The metal to be melted is placed in side the coil. The eddy currents generated in the metals produce heat, that melt it.

4. Electric power meters:
The metal disc in the electric power meter (analogue type) rotates due to the eddy currents. This rotation can be used to measure power consumption.

Inductance
An electric current can be induced in a coil by two methods:

1. Mutual induction
2. Self-induction

1. Mutual inductance:
Mutual induction:
The phenomenon of production of an opposing e.m.f. in a circuit due to the change in current or magnetic flux linked with a neighboring circuit is called mutual induction.
Explanation

Consider two coils P and S. P is connected to a battery and key. S is connected to a galvanometer. When the key is pressed, a change in magnetic flux is produced in the primary.

This flux is passed through S. So an e.m.f. is produced in the ‘S’. Thus we get a deflection in galvanometer. similarly, when key is opened, the galvanometer shows a deflection in the opposite direction.

Mutual inductance or coefficient of mutual induction:
The flux linked with the secondary coil is directly proportional to the current in the primary
i.e. Φ α I Or Φ = MI
Where M is called coefficient of mutual induction or mutual inductance.
If I = 1, M = Φ
Hence mutual inductance of two coils is numerically equal to the magnetic flux linked with one coil, when unit current flows through the other.

(i) Mutual inductance of two coils:
Expression for mutual inductance:
Consider a solenoid (air core) of cross-sectional area A and number of turns per unit length n. Another coil of total number of turns N is closely wound over the first coil. Let I be the current flow through the primary. Flux density of the first coil B= µ₀nI
Flux linked with second coil, Φ = BAN
Φ = µ₀nIAN _____(1)
But we know Φ = MI _______(2)
From eq(1) and eq(2), we get
∴ MI = µ₀nIAN
M = µ₀nAN
If the solenoid is covered over core of relative permeability µ_r
then M = µ_rµ₀nAN

Relation between induced e.m.f. and coefficient of mutual inductance:
Relation between induced e.m.f and mutual inductance
We know induced e.m.f

2. Self-inductance:
Self-induction
The phenomenon of production of an induced e.m.f in a circuit when the current through it changes is known as self-induction.
Explanation

Consider a coil connected to a battery and a key. When key is pressed, current is increased from zero to maximum value. This varying current produces a changing magnetic flux around the coil. The coil is situated in this changing flux, so that an e.m.f. is produced in the coil.

This induced e.m.f. is produced in the coil. This induced e.m.f is opposite to applied e.m.f (E). Hence this induced e.m.f is called back e.m.f.

Similarly, when key is released, a back e.m.f is produced which opposes the decay of current in the circuit.

Thus both the growth and decay of currents in a circuit is opposed by the back e.m.f. This phenomenon is called self – induction.

Mathematical expression for self-inductance:
Consider a solenoid (air core) of length /, number of turns N and area cross-section A. let ‘n’ be the no. of turns per unit length (n = N/l)
The magnetic flux linked with the solenoid,
Φ = BAN
Φ = µ₀nIAN (since B = µ₀nI)
but Φ = LI
∴ LI = µ₀nIAN
L = µ₀nAN
If solenoid contains a core of relative permeability µ_r the L = µ₀µ_rnAN.

Definition of self-inductance:
We know NΦ = LI
If I = 1, we get L = NΦ
Self-inductance (or) coefficient of self-induction may be defined as the flux linked with a coil, when a unit current is flowing through it.
Note: Physically, self-inductance plays the role of inertia.
The relation between induced emf and coefficient of self-induction:
When the current through a coil is varied, a back emf produced in the coil. Using Lens law, emf can be written as,

4. Energy stored in an inductor:
When the current in the coil is switched on, a back emf (ε = -Ldt/dt) is produced. This back emf opposes the growth of current. Hence work should be done, against this e.m.f.
Let the current at any instant be ‘I’ and induced emf
E = \(\frac{-\mathrm{d} \phi}{\mathrm{dt}}\)
i.e., work done, dw= EIdt

Hence the total work done (when the current grows from 0 to I₀ is)

This work is stored as potential energy.
V = \(\frac{1}{2}\)LI²

Ac Generator
An ac generator works on electromagnetic induction. AC generator converts mechanical energy into electrical energy.

The structure of an ac generator is shown in the above figure. It consists of a coil. This coil is known as armature coil. This coil is placed in between magnets. As the coil rotates, the magnetic flux through the coil changes. Hence an e.m.f. is induced in the coil.

1. Expression for induced emf:

Take the area of coil as A and magnetic field produced by the magnet as B. Let the coil be rotating about an axis with an angular velocity ω.

Let θ be the angle made by the areal vector with the magnetic field B. The magnetic flux linked with the coil can be written as

Φ = BA cosθ
Φ = BA cosωt [since θ = ωt)
If there are N turns
Φ = NBA cosωt
∴ The induced e.m.f. in the coil,

Let ε₀ = NAB ω,
then s = ε₀ sin ωt.

Expression for current:
When this emf is applied to an external circuit .alternating current is produced. The current at any instant is given by
I = \(\frac{V}{R}\)

(V = ε₀ sin ω)
I = I₀ sin ωt
Where I₀ = ε₀/R, it gives maximum value of current. The direction of current is changed periodically and hence the current is called alternating current.

Variation of AC voltage with time:

We hope the Plus Two Physics Notes Chapter 6 Electromagnetic Induction help you. If you have any query regarding Plus Two Physics Notes Chapter 6 Electromagnetic Induction, drop a comment below and we will get back to you at the earliest.

Plus Two Physics Notes Chapter 5 Magnetism and Matter is part of Plus Two Physics Notes. Here we have given Plus Two Physics Notes Chapter 5 Magnetism and Matter.

Kerala Plus Two Physics Notes Chapter 5 Magnetism and Matter

Introduction
The word magnet is derived from the name of an island in Greece called magnesia where magnetic ore deposits were found.
Properties of a magnet

1. When a bar magnet is freely suspended, it points in the north-south direction.
2. There is a repulsive force when north poles (or south poles) are brought close together.
3. We cannot isolate the north or south pole of a magnet.
4. It is possible to make magnets out of iron and its alloys.

Note: The earth behaves as a magnet with the magnetic field pointing approximately from the geographic south to the north.

The Bar Magnet
A magnet has two poles. One pole is North pole and the other South pole.
Magnetic poles:
These two points near the ends of a magnet at which the power of attraction of the magnet is mostly concentrated are called its magnetic poles.
Note: A current carrying solenoid behaves like a bar magnet.

1. The magnetic field lines:
Properties of magnetic field lines

1. The magnetic field lines of a magnet form continuous closed loops.
2. The tangent to the field line at a given point represents the direction of magnetic field at that point.
3. Flux density of magnetic field represents the strength of magnetic field.
4. The magnetic field lines do not intersect

Field lines of bar magnet:

Field lines of current-carrying solenoid:

Field lines of electric dipole:

2. Bar magnet as an equivalent solenoid:
Magnetic field along the axis of a solenoid or bar magnet

Consider a solenoid of radius ‘a’ and numberof turns per unit length‘n’. Let 2l be the length and I be the current flowing through the solenoid. Consider a point P at a distance ‘r’ from the centre of solenoid. To find magnetic field at P, we take a circular element of thickness dx at a distance x from the centre of solenoid.
The magnetic field at P due to this small element,

where ndx = N
(total number of turns in a circular element of thickness dx.)
Integrating from x = – l to x = + l, we get

If the point lies at large distance from the solenoid, we can take,
[(r – x)² + a²]^3/2 ≈ r³
r>>a, r>>x
Hence eq.(1) can be written as

Where m is called magnetic moment of the solenoid.

3. The dipole in a uniform magnetic field:
Torque acting on a magnetic dipole:
Consider a magnetic dipole of dipole moment ‘m’ placed in a uniform magnetic field B. If this dipole is rotated to an angle θ, a restoring torque will act on the needle. ie τ = -mBSinθ.
But we know rotational torque τ = la, where I is the moment of inertia of the magnetic dipole and α is the angular acceleration,
lα = -mBSinθ
(Restoring torque and rotational torque are equal in magnitude but opposite in direction)
But α = \(\frac{d^{2} \theta}{d t^{2}}\), for small rotations sinθ ≈ θ.

This equation represents that, the oscillation of this magnetic needle is simple harmonic. When we compare the above equation with standard harmonic.

Potential energy of a magnetic dipole:
The work done in rotating a magnet in a magnetic field is stored in it as its potential energy. If dipole is rotated through an angle (dθ) in a uniform magnetic field B, work done for this rotation,
dw = τdθ
dw = mBsinθdθ
If this magnetic needle is rotated from θ₁ to θ₂, total work done

If dipole is rotated from stable equilibrium (θ₁ = π/2) to θ₂ = 0 we get,
W = -mBcosθ

This work done is stored as magnetic potential energy, ie.

4. The electrostatic analog:
Permanent Magnets And Electromagnets

Substances that retain their ferromagnetic property at room temperature for a long time, even after the magnetizing field has been removed are called permanent magnets.

The hysteresis curve helps us to select such materials. They should have high retentivity so that the magnet is strong and high coercivity so that the magnetization is not lost by strong magnetic fields. The material should have a wide hysteresis loop. Steel, Alnico, cobalt-steel and nickel are examples.

Electromagnets are usually ferromagnetic materials with low retentivity, low coercivity and high permeability. The hysteresis curve should be narrow so that the energy liberated as heat is small.
The hysteresis curves of both these materials are shown in the above figure.

Magnetism And Gauss’s Law
Gauss’s law in magnetism: The net magnetic flux through any closed surface is zero.

Explanation:

Consider Gaussian surfaces represented by I and II. Both cases demonstrates that the number of magnetic field lines leaving the surface is balanced by the number of lines entering it. This is true for any closed surface.

The Earth’s Magnetism
Earth’s magnetic field rise due to electrical currents produced by motion of metallic fluids in the outer core of the earth. This is known as the dynamo effect.

The magnetic field of the earth behaves as magnetic dipole located at the centre of the earth. The axis of the dipole does not coincide with the axis of rotation of the earth. The axis of dipole is titled by 11.3° with axis of rotation of the earth.

The pole near the geographic north pole of the earth is called north magnetic pole (N_m). Likewise, the pole near the geographic south pole is called the south magnetic pole. (S_m).
Note: The north magnetic pole (N_m) behaves like the south pole of a bar magnet (inside the earth). Similarly, the south magnetic pole (S_m) behaves like the north pole of a bar magnet.

(i) Magnetic declination and dip:
The elements of earth’s magnetic field:
The earth’s magnetic field at a place can be completely specified in terms of three quantities. They are

1. Declination
2. Dip
3. Horizontal intensity

Magnetic meridian:
Magnetic meridian at a place is the vertical plane passing through the earth’s magnetic poles.
Geographic meridian:
Geographic meridian at a place is the vertical plane passing through the geographic poles.

1. Magnetic Declination (I):

Declination at a place is the angle between the geographic meridian and magnetic meridian at that place.

2. Dip or Inclination (θ):

The angle between the earth’s magnetic field and the horizontal component of the earth’s magnetic field at a place is called dip. Dip angle changes from place to place. On the equator, the dip is zero and at the poles, the dip is 90°.

3. Horizontal Intensity B_h:
The horizontal intensity at a place is the horizontal components of the earths field.
Relation between Dip, Horizontal intensity and Earth’s magnetic field:

Let B be the Earth’s magnetic field and θ be the angle of dip. Let B_h be the horizontal intensity and Bvthe vertical intensity of the earth’s magnetic field. Then from figure, we get
B_h = B cos θ
The vertical component, B_v = B sin θ
∴ Tanθ = \(\frac{B_{v}}{B_{h}}\)
and resultant field, B = \(\sqrt{\mathbf{B}_{\mathrm{h}}^{2}+\mathbf{B}_{\mathrm{v}}^{2}}\)

Magnetization And Magnetic Intensity
The magnetic properties of a substance can be studied by defining some parameter such as

1. Intensity of magnetization (M)
2. Magnetic intensity vector (H)
3. Susceptibility
4. Permeability

1. Intensity of magnetisation (M):
It is defined as the magnetic moment per unit volume. It is the measure of the extent to which a specimen is magnetized.

2. Magnetic Intensity Vector (Magnetising field):
It is defined as the magnetic field which produces an induced magnetism in a magnetic substance. If H is the magnetising field and B the induced magnetic field in the material.
ie. H = \(\frac{B}{\mu}\)
where µ is the constant called the magnetic permeability of the medium.

3. Magnetic susceptibility (χ):
Magnetic susceptibility of a specimen is the ratio of its magnetization to the magnetising field,
ie. χ = \(\frac{M}{H}\)

4. Magnetic permeability (µ):
It is the ratio of magnetic field inside a specimen to the magnetising field.
ie. µ = \(\frac{B}{H}\)
µ = µ₀µ_r
µ₀ – Permeability of free space
µ_r – Relative permeability of a medium.

Relation between permeability and susceptibility:
Let a magnetic material be kept in a solenoid. The specimen gets magnetized by induction. The resultant field inside the specimen is the sum of the field due to the current in the solenoid and the field due to the magnetization of the material.
Resultant field B = Field due to current B₀ + Field due to magnetization B_m.
∴ B = B₀ + B_m
But B_m = µ₀M, B₀ = µ₀H
∴ B = µ₀H + µ₀M

Magnetic Properties Of Materials
Materials can be classified as diamagnetic, paramagnetic or ferromagnetic in terms of the susceptibility χ. A material is diamagnetic if χ is negative, para-if χ is positive and small, and Ferro-if χ is large and positive.

1. Diamagnetism:
Diamagnetic substances are those which have tendency to move from stronger to the weaker part of the external magnetic field.
Diamagnetic material in external magnetic field:

Figure shows a bar of diamagnetic material placed in an external magnetic field. The field lines are repelled and the field inside the material is reduced.

Explanation of diamagnetism:
Electrons in an atom orbit around nucleus. These orbiting electrons produce magnetic field. Hence atom possess magnetic moment.

Diamagnetic substances are the ones in which resultant magnetic moment in an atom is zero. When magnetic field is applied, those electrons having orbital magnetic moment in the same direction slow down and those in the opposite direction speed up.

Thus, the substance develops a net magnetic moment in direction opposite to that of the applied field and hence it repels external magnetic field.

Examples:
Some diamagnetic materials are bismuth, copper, lead, silicon, nitrogen (at STP), water and sodium chloride.

Meissner effect:
The phenomenon of perfect diamagnetism in superconductors is called the Meissner effect.

2. Paramagnetism:
Paramagnetic substances are those which get weakly magnetized in an external magnetic field. They get weakly attracted to a magnet.

Reason for paramagnetism:
The atoms of a paramagnetic material possess a permanent magnetic dipole moment. But these magnetic moments are arranged in all directions.

Due to this random arrangement net magnetic moment becomes zero. But in the presence of an external field B₀, the atomic dipole moment can be made to align in the same direction of B0. Hence paramagnetic material shows magnetism in external magnetic field.
Paramagnetic material in external magnetic field:

Figure shows a bar of paramagnetic material placed in an external field. The field lines gets concentrated inside the material, and the field inside is increased.

Examples:
Some paramagnetic materials are aluminium, sodium, calcium, oxygen (at STP) and copper chloride.

Curie law of magnetism:
Curie law of magnetism states that the magnetisation of a paramagnetic material is inversely proportional to the absolute temperature T.

In the case of paramagnetic materials it can be shown that the magnetic susceptibility at temperature is given by

Where C is a constant called curie’s constant.

3. Ferromagnetism:
Ferromagnetic substances are those which gets strongly magnetized in an external magnetic field. They get strongly attracted to a magnet.
Ferro magnetic materials without external magnetic field:

The atoms in a ferromagnetic material possess a dipole moment. These dipoles align in a common direction over a macroscopic volume called domain. Each domain has a net magnetization. Domains are arranged randomly. Hence net magnetic moment of all domains is zero, so ferromagnetic substance does not show magnetism.
Ferro magnetic materials in external magnetic field:

When we apply an external magnetic field B₀, the domains arranged in the direction of B₀ and grow in size. Thus, in a ferromagnetic material the field lines are highly concentrated.

Question 1.
What happens when the external field is removed?
Answer:
In some ferromagnetic materials the magnetisation persists even if external field is removed. Such materials are called hard ferromagnets. Such materials are used to make permanent magnets.
Eg: Alnico
There is a another class of ferromagnetic materials in which the magnetisation disappears on removal of the external field. Such materials are called soft ferromagnetic materials.
Eg: soft iron

Curie temperature:
The ferromagnetic property depends on temperature. At high temperature, a ferromagriet becomes a paramagnet. The domain structure disintegrates with temperature. This disappearance of magnetisation with temperature is gradual. The temperature of transition from ferromagnetic to paramagnetism is called the Curie temperature T_c.

Variation of B and H in paramagnetic materials:
Figure shows the plot of B versus H. As H is gradually increased from zero, B also increase from zero along OP.

As H increases, more and more magnetic dipoles get aligned in the direction of the field. So M increases and hence B increases.

When all the dipoles get aligned in the direction of the field, the curve becomes almost flat. After this there is no increase of B with H.

When H is gradually decreased from P₁ there is no corresponding decrease in the magnetization. The shifting of domains in the ferromagnetic materials is not completely reversible and some magnetization remains even when H is reduced to zero.

The value of the magnetic field when H is zero is called the remanent field Br (Retentivity). If the current in the solenoid is now reversed so that H is in the opposite direction, the magnetic field B can be gradually brought to zero at the point C. The value of H needed to reduce B to zero is called the coercive force He (Coercivity).

The remaining part of curve is obtained by applying H in reverse direction. From these variations it is clear that B always lags behind H. This phenomenon is known as magnetic hysteresis (hysteresis means to lag behind).

The area enclosed by the hysteresis curve gives the loss of energy in the form of heat during the magnetisation – demagnetization cycle.

Permanent Magnets And Electromagnets

Substances which retain their ferromagnetic property at room temperature for a longtime, even after the magnetizing field has been removed are called permanent magnets.

The hysteresis curve helps us to select such materials. They should have high retentivity so that the magnet is strong and high coercivity so that the magnetization is not lost by strong magnetic fields. The material should have a wide hysteresis loop. Steel, Alnico, cobalt-steel and nickel are examples.

Electromagnets are usually ferromagnetic materials with low retentivity, low coercivity and high permeability. The hysteresis curve should be narrow so that the energy liberated as heat is small.
The hysteresis curves of both these materials are shown in the above figure.

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