How do you calculate the total resistance of a parallel circuit?

How do you calculate the total resistance of a parallel circuit?

The Effective Resistance of Resistors Connected in Parallel

  1. There are three important characteristics in a parallel circuit:
    (a) The potential difference is the same across each resistor.
    (b) The current that passes through each resistor is inversely proportional to the resistance of the resistor.
    (c) The total current in the circuit equals to the sum of the currents passing through the resistors in its parallel branches.
    parallel combination of resistance 1
  2. When two or more resistances are connected between two common points so that the same potential difference is applied across each of them, they are said to be connected is parallel.
    How do you calculate the total resistance of a parallel circuit 1When such a combination of resistance is connected to a battery, all the resistances have the same potential difference across their ends.
  3. Derivation of mathematical expression of parallel combination:
    Let, V be the potential difference across the two common points A and B. Then, from Ohm’s law
    Current passing through R1,     I1 = V/R1             … (i)
    Current passing through R2,     I2 = V/R2           … (ii)
    Current passing through R3,     I3 = V/R3          … (iii)
  4. If R is the equivalent resistance, then from Ohm’s law, the total current flowing through the circuit is given by,
    I = V/R                               … (iv)
    and I = I1 + I2 + I3           … (v)
  5. Substituting the values of I, I1, I2 and I3 in Eq. (v),
    \( \frac{\text{V}}{\text{R}}=\frac{\text{V}}{{{\text{R}}_{\text{1}}}}+\frac{\text{V}}{{{\text{R}}_{\text{2}}}}+\frac{\text{V}}{{{\text{R}}_{\text{3}}}}\text{ }……..\text{ (vi)} \)
  6. Cancelling common V term, one gets
    \( \frac{\text{1}}{\text{R}}=\frac{\text{1}}{{{\text{R}}_{\text{1}}}}+\frac{\text{1}}{{{\text{R}}_{2}}}+\frac{\text{1}}{{{\text{R}}_{3}}} \)
    The equivalent resistance of a parallel combination of resistance is less than each of all the individual resistances.
  7. The equivalent circuit is shown in Figure.parallel combination of resistance 2

Important results about parallel combination:

  1.  Total current through the circuit is equal to the sum of the currents flowing through it.
  2.  In a parallel combination of resistors the voltage (or potential difference) across each resistor is the same and is equal to the applied voltage i.e. V1 = V2 = V3 = V.
  3.  Current flowing through each resistor is inversely proportional to its resistances, thus higher the resistance of a resistors, lower will be the current flowing through it.

People also ask

Parallel Circuit Problems with Solutions

  1. The three resistors, R1, Rand R3, in are connected in parallel to the battery as shown in Figure.
    Parallel Circuit Problems with Solutions 2
    Calculate
    (a) the potential difference across each resistor,
    (b) the effective resistance, R of the circuit,
    (c) the current, I, in the circuit,
    (d) the currents, I1, I2 and I3 passing through each resistor.
    Solution:
    (a) Since this is a parallel circuit, the potential difference across each resistor is 6 V, same as the potential difference across the battery, which is 6 V.
    Parallel Circuit Problems with Solutions 1

How do you calculate the total resistance of a series circuit?

How do you calculate the total resistance of a series circuit?

The Effective Resistance of Resistors Connected in Series

  1. There are three important characteristics in a series circuit:
    (a) The current passing through each resistor is the same.
    (b) The potential difference across each resistor depends directly on its resistance.
    (c) The sum of the potential difference across each resistor is equal to the total potential difference of the source.
    series combination of resistances 1
  2. When two or more resistances are joined end-to-end so that the same current flows through each of them, they are said to be connected in series. When a series combination of resistances is connected to a battery, the same current (I) flows through each of them.
    How do you calculate the total resistance of a series circuit 1When a series combination of resistances is connected to a battery, the same current (I) flows through each of them.
  3. Law of combination of resistances in series: The law of combination of resistances in series states that when a number of resistances are connected in series, their equivalent resistance is equal to the sum of the individual resistances. Thus, if R1, R2, R3 …, etc. are combined in series, then the equivalent resistance (R) is given by,
    R = R1 + R2 + R3 + …                          …. (i)
  4. Derivation of mathematical expression of resistances in series combination:
    Let, R1, Rand R3 be the resistances connected in series, I be the current flowing through the circuit, i.e., passing through each resistance, and V1, Vand V3 be the potential difference across R1, Rand R3 respectively. Then, from Ohm’s law,
    V1 = IR1, V2 = IR2 and V3 = IR3      … (ii)
  5. If, V is the potential difference across the combination of resistances then,
    V = V1 + V2 + V3                                 … (iii)
  6. If, R is the equivalent resistance of the circuit, then
    V = IR                                                   … (iv)
  7. Using Equations (i) to (iv) we can write,
    IR = V = V1 + V2 + V3
    IR = IR1 + IR2 + IR3
    IR = I (R1 + R2 + R3 )
    R = R1 + R2 + R3
    Therefore, when resistances are combined in series, the equivalent resistance is higher than each individual resistance.
  8. The equivalent circuit is shown in Figure.
    series combination of resistances 2

Some results about series combination:

  1. When two or more resistors are connected in series, the total resistance of the combination is equal to the sum of all the individual resistances.
  2. When two or more resistors are connected in series, the same current flows through each resistor.
  3. When a number of resistors are connected in series, the voltage across the combination
    (i.e. voltage of the battery in the circuit), is equal to the sum of the voltage drop
    (or potential difference) across each individual resistor.

People also ask

Series Circuit Problems with Solutions

  1. Three resistors, R1, Rand R3, are connected in series to a 6 V battery as shown in Figure.
    Series Circuit Problems with Solutions
    Calculate
    (a) the effective resistance, R of the circuit,
    (b) the current, I in the circuit,
    (c) the potential differences across each resistor, V1, Vand V3.
    Solution:
    Series Circuit Problems with Solutions 1

    Note that the larger the resistance, the larger the potential difference across it. The sum of the potential difference across each resistor is the same as the potential difference across the battery.