Question

An AC generator producing $10\,{\text{V}}$ (rms) at ${\text{200}}\,{\text{rad/s}}$ is connected in series with a ${\text{50}}\,\Omega$ resistor, a $400\,{\text{mH}}$ inductor and a ${\text{200}}\,\mu{\text{ F}}$ capacitor. The rms voltage across the inductor is
A. $2.5\,{\text{V}}$
B. $3.4\,{\text{V}}$
C. $6.7\,{\text{V}}$
D. $10.8\,{\text{V}}$

Answer 1

We use the formula of capacitance reactance, inductive reactance, impedance and we also use the given information for this solution and solve it accordingly.

Complete step by step answer:
Given,
$E=10 ,{\text{V}}, \omega = 200, R = 50\,\Omega , L = 400,{\text{mH}}, C=200 {\text{F}}$

From capacitance reactance we get,
{X_C} = \dfrac{1}{{\omega C}} \\\ {X_C} = \dfrac{1}{{\left( {200} \right) \times \left( {200 \times {{10}^{ - 6}}} \right)}} \\\ {X_C} = 25\,\Omega \\\

So the inductive reactance will be,

{X_L} = \omega L \\\ {X_L} = \left( {200} \right)\left( {400 \times {{10}^{ - 3}}} \right) \\\ {X_L} = 80\,\Omega \\\
And impedance,

$Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \\\ Z = \sqrt {{{50}^2} + {{\left( {80 - 25} \right)}^2}} \\\ Z = 74.3\,\Omega \\\$
So,

I = \dfrac{E}{Z} \\\ I = \dfrac{{10}}{{74.3}} \\\ I = 0.13459\,{\text{A}} \\\
And

{E_L} = I{X_L} \\\ {E_L} = 0.1345 \times 80 \\\ {E_L} = 10.76\,{\text{V}} \\\

Hence the required answer is $10.76\,{\text{V}} \approx {\text{10}}{\text{.8}}\,{\text{V}}$.

So, the correct answer is “Option D”.

Additional Information:
Capacitance reactance: Capacitive reactance ($X_C$ symbol) is an anti-AC calculation of the capacitor. It is calculated in ohm like resistance, but reaction is complicated rather than resistance, since its value depends on the frequency (f) of the signal passing through the capacitor.

Inductive reactance: The name given to the resistance to a change in current flow is an inductive reaction. The impedance is just like resistance calculated in ohm. The voltage of the inductors contributes to a $90$-degree current.

Impedance: Electrical impedance is the calculation of the opposition of a circuit to a current in the application of stress. The impedance of a two-terminal circuit element is quantitatively a ratio between its terminals, between the complex representation of the sinusoidal voltage and the complex representation of the current that flows through it. It depends generally on the sine voltage frequency. Impedance applies the resistance principle to alternating current (AC) circuits and has magnitude and phase, unlike magnitude resistance. There is no difference between impedance and durability when a circuit is operated by direct current (DC).

Note:
Capacitive reaction, by comparison, is related to a shift in electric field between two leading surfaces (plates) separated by an isolating medium from each other. Such a number of conductors, a capacitor, primarily counteract changes in voltage or possible difference on their surfaces.