Question

The same current is flowing in two alternating circuits. The first circuit contains only inductance and the other contains only a capacitor. If the frequency of the emf of ac is increased the effect on the value of the current will be.

• A. Increase in the first circuit and decrease in other
• B. Increase in both the circuit
• C. Decrease in both the circuit
• D. Decrease in the first and increase in other

Answer 1

The correct option is D

Decrease in the first and increase in other

Explanation for the correct option:

Step 1: Given Data

1. Given, that the current in the two conductors is the same. Let $I$ denote the current flowing.
2. The first conductor contains only inductance. Let the inductance be denoted by $L$.
3. The second conductor only contains capacitance. Let the capacitance be denoted by $C$.

Step 2: Formulas used

According to Ohm's law,
$\begin{array}{cc}& V=IZ\\ \Rightarrow & I=\frac{V}{Z}\end{array}$
where $V$ is the voltage, $I$ is the current and $Z$ is the impedance.

We also know that,
$Z=\sqrt{R^2+{\left(X_L-X_C\right)}^2}$
Where $R$ is the resistance, $X_L$ is the inductive reactance and $X_C$ is the capacitive reactance,

We also know,

The inductive reactance and the capacitive reactance in terms of frequency of current is,
$$\begin{gathered} X_L=\omega L \\ {X}_c=\frac{1}{\omega C} \end{gathered}$$
where $\omega$ is the frequency of the AC current.

Step 3: Solution

For the first circuit (only the inductor is in the circuit) $X_c=0$

So current in it is,
$\begin{array}{rcl}I& =& \frac{V}{\sqrt{R^2+{X_L}^2}}\\ & =& \frac{V}{\sqrt{R^2+{\omega }^2{L}^2}}\end{array}$

We can observe that as frequency ($\omega$) increases, inductive reactance increases so impedance also increases which implies that current decreases.

For the second circuit (only the capacitor is in the circuit) $X_L=0$

So current in it is,
$\begin{array}{rcl}I& =& \frac{V}{\sqrt{R^2+{X_C}^2}}\\ & =& \frac{V}{\sqrt{R^2+{\displaystyle \frac{1}{{\omega }^2{C}^2}}}}\end{array}$

We can observe that as frequency ($\omega$) increases, capacitive reactance decreases so impedance also decreases which implies that current increases.

Hence, option D is correct.