Question

The values of ${X_L}$, ${X_C}$ and $R$ in an AC circuit are $8\,\Omega$, $6\,\Omega$ and $10\,\Omega$ respectively. The total impedance of the circuit:
(A) $10.2\,\Omega$
(B) $12.2\,\Omega$
(C) $10\,\Omega$
(D) $24.4\,\Omega$

Answer 1

The impedance of the circuit is determined by using the impedance of the circuit formula, by using this formula and also by using the values of the inductive reactance, capacitive reactance and the resistance values, then the impedance in the circuit can be determined.

Formula used:
The impedance of the circuit is given by,
$Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}}$
Where, $Z$ is the impedance of the circuit, ${X_L}$ is the inductive reactance of the circuit, ${X_C}$ is the capacitive reactance of the circuit and $R$ is the resistance of the circuit.

Complete step by step solution:
Given that,
The inductive reactance of the circuit is, ${X_L} = 8\,\Omega$
The capacitive reactance of the circuit is, ${X_C} = 6\,\Omega$
The resistance of the circuit is, $R = 10\,\Omega$
Now,
The impedance of the circuit is given by,
$Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \,................\left( 1 \right)$
By substituting the inductive reactance of the circuit, the capacitive reactance of the circuit and the resistance of the circuit in the above equation (1), then the above equation (1) is written as,
$Z = \sqrt {{{10}^2} + {{\left( {8 - 6} \right)}^2}}$
By subtracting the terms in the above equation, then the above equation is written as,
$Z = \sqrt {{{10}^2} + {2^2}}$
By squaring the terms inside the square root in the above equation, then the above equation is written as,
$Z = \sqrt {100 + 4}$
By adding the terms inside the square root in the above equation, then the above equation is written as,
$Z = \sqrt {104}$
By taking the square root in the above equation, then the above equation is written as,
$Z = 10.19\,\Omega$
Then the above equation is approximately written as,
$Z \simeq 10.2\,\Omega$

Hence, the option (A) is the correct answer.

Note: The impedance of the circuit is dependent only on the resistance of the circuit, the inductive reactance of the circuit, the capacitive reactance of the circuit. The impedance is also the form of the resistance and it is the measure of the overall opposition of the current in the circuit.