Prasanna

Ohm’s Law Definition – Formula and its Current Electricity

The ohm’s law can be derived from the equation J = σE. Consider a segment of wire of length l and cross sectional area A as shown in Figure 2.7

When a potential difference V is applied across the wire, a net electric field is created in the wire which constitutes the current in the wire. For simplicity, we assume that the electric field is uniform in the entire length of the wire, then the potential difference (voltage V) can be written as

V = El

As we know, the magnitude of current density
J = σE = σ \(\frac{V}{l}\) ………… (2.14)

But J = \(\frac{I}{A}\), so we write the equation (2.14) as
\(\frac{I}{A}\) = σ \(\frac{V}{l}\).

By rearranging the above equation, we get
V = I(\(\frac{1}{σA}\)) ………….. (2.15)

The quantity \(\frac{1}{σA}\) is called resistance of the conductor and it is denoted as R. Note that the resistance is directly proportional to the length of the conductor and inversely proportional to area of cross section.

Therefore, the macroscopic form of ohm’s law can be stated as

V = IR …………… (2.16)

From the above equation, the resistance is the ratio of potential difference across the given conductor to the current passing through the conductor.

R = \(\frac{V}{I}\) ………………. (2.17)

The SI unit of resistance is ohm (Ω). From the equation (2.16), we infer that the graph between current versus voltage is straight line with a slope equal to the inverse of resistance R of the conductor. It is shown in the Figure 2.8 (a).

Materials for which the current versus voltage graph is a straight line through the origin, are said to obey Ohm’s law and their behaviour is said to be ohmic as shown in Figure 2.8(a). Materials or devices that do not follow Ohm’s law are said to be nonohmic. These materials have more complex relationships between voltage and current. A plot of I versus V for a non-ohmic material is non-linear and they do not have a constant resistance (Figure 2.8(b)).

Example 2.5

A potential difference across 24 Ω resistor is 12 V. What is the current through the resistor?

Solution

V = 12 V and R = 24 Ω
Current, I = ?
From Ohm’s law, I = \(\frac{V}{R}\) = \(\frac{12}{24}\) = 0.5A

Resistivity

In the previous section, we have seen that the resistance R of any conductor is given by

R = \(\frac{1}{σA}\) …………… (2.18)

where σ is called the conductivity of the material and it depends only on the type of the material used and not on its dimension. The resistivity of a material is equal to the reciprocal of its conductivity.

ρ = \(\frac{1}{σ}\) ……….. (2.19)

Now we can rewrite equation (2.18) using equation (2.19)

R = ρ\(\frac{l}{A}\) …………. (2.20)

The resistance of a material is directly proportional to the length of the conductor and inversely proportional to the area of cross section of the conductor. The proportionality constant ρ is called the resistivity of the material.

If l = 1 m and A = 1 m², then the resistance R = ρ. In other words, the electrical resistivity of a material is defined as the resistance offered to current flow by a conductor of unit length having unit area of cross section. The SI unit of ρ is ohm-metre (Ω m).

Based on the resistivity, materials are classified as conductors, insulators and semiconductors. The conductors have lowest resistivity, insulators have highest resistivity and semiconductors have resistivity greater than conductors but less than insulators. The typical resistivity values of some conductors, insulators and semiconductors are given in the Table 2.1

Example 2.6

The resistance of a wire is 20 Ω. What will be new resistance, if it is stretched uniformly 8 times its original length?

Solution

R₁ = 20 Ω, R₂ = ?
Let the original length of the wire (l₁) be l.
New length, l₂ = 8l₁ (i.e;) l₂ = 8l
Original resistance, R₁ = ρ\(\frac{l_{1}}{A_{1}}\)
New resistance R₂ = ρ\(\frac{l_{2}}{A_{2}}\) = \(\frac{\rho(8 l)}{A_{2}}\)
Though the wire is stretched, its volume remains unchanged.
Initial volume = Final volume
A₁l₁ = A₂l₂, A₁l = A₂(8l)
\(\frac{A_{1}}{A_{2}}\) = \(\frac{8l}{l}\) = 8
By dividing equation for R₂ by equation for R₁, we get

Substituting the value of \(\frac{A_{1}}{A_{2}}\), we get
\(\frac{R_{2}}{R_{1}}\) = 8 × 8 = 64
R₂ = 64 × 20 = 1280 Ω

Hence, stretching the length of the wire has increased its resistance

Example 2.7

Consider a rectangular block of metal of height A, width B and length C as shown in the figure.

If a potential difference of V is applied between the two faces A and B of the block (figure (a)), the current I_AB is observed. Find the current that flows if the same potential difference V is applied between the two faces B and C of the block (figure (b)). Give your answers in terms of I_AB.

Solution

In the first case, the resistance of the block

R_AB = ρ\(\frac{length}{Area}\) = ρ\(\frac{C}{AB}\)
The current I_AB = \(\frac{V}{R_{A B}}\) = \(\frac{V}{ρ}\).\(\frac{AB}{C}\) ………. (1)

In the second case, the resistance of the block R_BC = ρ\(\frac{A}{BC}\)
The current I_BC = \(\frac{V}{R_{B C}}\) = \(\frac{V}{\rho}\). \(\frac{BC}{A \) ………… (2)

To express I_BC interms of I_AB, we multiply and divide equation (2) by AC, we get I_BC = \(\frac{V}{ρ}\). \(\frac{BC}{A}\)\(\frac{AC}{AC}\) = (\(\frac{V}{ρ}\) \(\frac{AB}{C}\)).\(\frac{C^{2}}{A^{2}}\) = \(\frac{C^{2}}{A^{2}}\).I_AB Since C > A, the current I_BC > I_AB

Resistors in Series and Parallel

An electric circuit may contain a number of resistors which can be connected in different ways. For each type of circuit, we can calculate the equivalent resistance produced by a group of individual resistors.

Resistors in Series

When two or more resistors are connected end to end, they are said to be in series. The resistors could be simple resistors or bulbs or heating elements or other devices. Figure 2.9 (a) shows three resistors R₁, R₂ and R₃ connected in series.

The amount of charge passing through resistor R₁ must also pass through resistors R₂ and R₃ since the charges cannot accumulate anywhere in the circuit.

Due to this reason, the current I passing through all the three resistors is the same. According to Ohm’s law, if same current pass through different resistors of different values, then the potential difference across each resistor must be different.

If V₁, V₂ and V₃ be the potential differences (voltage) across each of the resistors R₁, R₂ and R₃ respectively, then we can write V₁ = IR₁, V₂ = IR₂ and V₃ = IR₃. But the supply voltage V must be equal to the sum of voltages (potential differences) across each resistor.

V = V₁ + V₂ + V₃ = IR₁ + IR₂ + IR₃ ……… (2.21)
V = I (R₁ + R₂ + R₃)
V = IR_S ………… (2.22)

where R_s is the equivalent resistance.

R_S = R₁ + R₂ + R₃ ………….. (2.23)

When several resistors are connected in series, the total or equivalent resistance is the sum of the individual resistances as shown in the Figure 2.9 (b).

Note:
The value of equivalent resistance in series connection will be greater than each individual resistance.

Example 2.8

Calculate the equivalent resistance for the circuit which is connected to 24 V battery and also find the potential difference across each resistors in the circuit.

Solution

Since the resistors are connected in series, the effective resistance in the circuit
= 4 Ω + 6 Ω = 10 Ω
current I in the circuit = \(\frac{V}{R_{e q}}\) = \(\frac{24}{10}\) = 2.4A
Voltage across 4Ω resistor
V₁ = IR₁ = 2.4A × 4Ω = 9.6V
Voltage across 6Ω resistor
V₂ = IR₂ = 2.4A × 6Ω = 14.4V

Resistors in Parallel

Resistors are in parallel when they are connected across the same potential difference as shown in Figure 2.10 (a).

In this case, the total current I that leaves the battery is split into three separate components. Let I₁, I₂ and I₃ be the current through the resistors R₁, R₂ and R₃ respectively. Due to the conservation of charge, total current in the circuit I is equal to sum of the currents through each of the three resistors.

I = I₁ + I₂ + I₃ ……….. (2.24)

Since the voltage across each resistor is the same, applying Ohm’s law to each resistor, we have

Substituting these values in equation (2.24), we get

Here R_p is the equivalent resistance of the parallel combination of the resistors. Thus, when a number of resistors are connected in parallel, the sum of the reciprocals of resistance of the individual resistors is equal to the reciprocal of the effective resistance of the combination as shown in the Figure 2.10 (b).

Note:
The value of equivalent resistance in parallel connection will be lesser than each individual resistance.

House hold appliances are always connected in parallel so that even if one is switched off, the other devices could function properly.

Example 2.9

Calculate the equivalent resistance in the following circuit and also find the values of current I, I₁ and I₂ in the given circuit.

Solution

Since the resistances are connected in parallel, the equivalent resistance in the circuit is

The resistors are connected in parallel, the potential difference (voltage) across them is the same.

The current I is the sum of the currents in the two branches. Then,
I = I₁ + I₂ = 6A + 4A = 10 A

Example 2.10

Two resistors when connected in series and parallel, their equivalent resistances are 15 Ω and \(\frac{56}{15}\)Ω respectively. Find the values of the resistances.

Solution

R_s = R₁ + R₂ = 15 Ω ……….. (1)
R_p = \(\frac{R_{1} R_{2}}{R_{1}+R_{2}}\) = \(\frac{56}{15}\)Ω
∴ R₁R₂ = 56
R₂ – \(\frac{56}{R_{1}}\) Ω ………….. (2)
From equation (1) substituting for R₁ + R₂ in equation (2)
\(\frac{R_{1} R_{2}}{15}\) = \(\frac{56}{15}\)Ω
∴ R₁R₂ = 56
R₂ = \(\frac{56}{R_{1}}\)Ω …………… (3)
Substituting for R₂ in equation (1) from equation (3)
R₁ + \(\frac{56}{R_{1}}\) = 15
Then, \(\frac{R_{1}^{2}+56}{R_{1}}\) = 15
R₁² + 56 = 15R₁
R₁² – 15R₁ + 56 = 0

The above equation can be solved using factorisation

R₁ = 8 Ω (or) R₁ = 7 Ω
If R₁ = 8 Ω
Substituting in equation (1)
8 + R₂ = 15
R₂ = 15 – 8 = 7 Ω
If R₂ = 7 Ω i.e; (when R₁ = 8 Ω ; R₂ = 7 Ω)
If R₁ = 7 Ω
Substituting in equation (1)
7 + R₂ = 15
R₂ = 8 Ω, i.e; (when R₁ = 7 Ω; R₂ = 8 Ω)

Example 2.11

Calculate the equivalent resistance between A and B in the given circuit.

Solution

In all the sections, the resistors are connected in parallel.

Equivalent resistance is given by

R = R_p1 + R_p2 + R_p3
R = 1Ω + 2Ω + 3Ω = 6Ω
The circuit becomes

Equivalent resistance between A and B is

Example 2.12

Five resistors are connected in the configuration as shown in the figure. Calculate the equivalent resistance between the points a and b.

Solution

Case (a)

To find the equivalent resistance between the points a and b, we assume that a current is entering the junction at a. Since all the resistances in the outside loop are the same (1Ω), the current in the branches ac and ad must be equal. Hence the points C and D are at the same potential and no current through 5 Ω. It implies that the 5 Ω has no role in determining the equivalent resistance and it can be removed. So the circuit is simplified as shown in the figure

The equivalent resistance of the circuit between a and b is R_eq = 1 Ω

Colour code for Carbon resistors

Carbon resistors consists of a ceramic core, on which a thin layer of crystalline carbon is deposited as shown in Figure 2.11. These resistors are inexpensive, stable and compact in size. Colour rings are used to indicate the value of the resistance according to the rules given in the Table 2.2.

Three coloured rings are used to indicate the values of a resistor: the first two rings are significant figures of resistances, the third ring indicates the decimal multiplier after them. The fourth colour, silver or gold,

shows the tolerance of the resistor at 10% or 5% as shown in the Figure 2.12. If there is no fourth ring, the tolerance is 20%.

For the resistor shown in Figure 2.12, the first digit = 5 (green), the second digit = 6 (blue), decimal multiplier = 10³ (orange) and tolerance = 5% (gold). The value of resistance = 56 × 10³ Ω or 56 kΩ with the tolerance value 5%.

Temperature dependence of resistivity

The resistivity of a material is dependent on temperature. It is experimentally found that for a wide range of temperatures, the resistivity of a conductor increases with increase in temperature according to the expression, ρ_r = ρ₀[1 + α(T – T₀)] ……….. (2.27)

where ρ_T is the resistivity of a conductor at T⁰C, ρ₀ is the resistivity of the conductor at some reference temperature T₀ (usually at 20°C) and α is the temperature coefficient of resistivity. It is defined as the ratio of increase in resistivity per degree rise in temperature to its resistivity at T₀.

From the equation (2.27), we can write

where ∆ρ = ρ_τ – ρ₀ is change in resistivity for a change in temperature ∆T = T – T₀. Its unit is per oC.

α of Conductors

For conductors α is positive. If the temperature of a conductor increases, the average kinetic energy of electrons in the conductor increases. This results in more frequent collisions and hence the resistivity increases. The graph of the equation (2.27) is shown in Figure 2.13.

Even though, the resistivity of conductors like metals varies linearly for wide range of temperatures, there also exists a nonlinear region at very low temperatures. The resistivity approaches some finite value as the temperature approaches absolute zero as shown in Figure 2.13(b).

Using the equation ρ = R \(\frac{A}{l}\) in equation (2.27), we get the expression for the resistance of a conductor at temperature T°C as

R_r = R₀[1 + α(T – T₀)] ……………. (2.28)

The temperature coefficient of resistivity can also be obtained from the equation (2.28),

where ∆R = R_T – R₀ is change in resistance during the change in temperature ∆T = T – T₀

α of Semiconductors

For semiconductors, the resistivity decreases with increase in temperature. As the temperature increases, more electrons will be liberated from their atoms (Refer unit 9 for conduction in semi conductors)

Hence the current increases and therefore the resistivity decreases as shown in Figure 2.14. A semiconductor with a negative temperature coefficient of resistivity is called a thermistor.

The typical values of temperature coefficients of various materials are given in table 2.3.

We can understand the temperature dependence of resistivity in the following way. In section 2.1.3, we have shown that the electrical conductivity, σ = \(\frac{n e^{2} \tau}{m}\). As the resistivity is inverse of σ, it can be written as ρ = \(\frac{m}{n e^{2} \tau}\) …………. (2.30)

The resistivity of materials is

(i) inversely proportional to the number density (n) of the electrons
(ii) inversely proportional to the average time between the collisions (τ).

In metals, if the temperature increases, the average time between the collision (τ) decreases and n is independent of temperature. In semiconductors when temperature increases, n increases and τ decreases, but increase in n is dominant than decreasing τ, so that overall resistivity decreases.

Example 2.3

If the resistance of coil is 3 Ω at 20° C and α = 0.004/°C then determine its resistance at 100 °C.

Solution

R₀ = 3 Ω, T = 100° C, T₀ = 20°C
α = 0.004/°C, R_T = ?
R_T = R₀(1 + α(T – T₀))
R₁₀₀ = 3(1 + 0.004 × 80)
R₁₀₀ = 3.96 Ω

Example 2.4

Resistance of a material at 20°C and 40°C are 45 Ω and 85 Ω respectively. Find its temperature coefficient of resistivity.

Solution

T₀ = 20°C, T = 40°C, R₀ = 45 Ω, R = 85 Ω
α = \(\frac{1}{R_{\circ}}\) \(\frac{∆R}{∆T}\)
α = \(\frac{1}{45}\)(\(\frac{85-45}{40-20}\)) = \(\frac{1}{45}\)(2)
α = 0.044 per°C

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