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Chapter 14: Problem 2083

Same current is flowing in two alternating circuits. The first circuits 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

Short Answer

Expert verified
(d) Decrease in the first and increase in the other

Step by step solution

01

Recall the formula for the impedance of an inductor and a capacitor

Impedance is the opposition offered by any circuit element to the flow of current. The impedance of an inductor (L) is given by \(Z_L = j\omega L\), where \(j\) is the imaginary unit, \(\omega\) is the angular frequency, and \(L\) is the inductance. The impedance of a capacitor (C) is given by \( Z_C = \frac{1}{j\omega C}\), where \(C\) is the capacitance.
02

Examine the effect of increasing frequency on the impedance

In an AC circuit, the angular frequency (\(\omega\)) is related to the frequency (f) through the formula \(\omega = 2\pi f\). Therefore, the impedances of the inductor and the capacitor become: \(Z_L = j(2\pi f L)\) and \( Z_C = \frac{1}{j(2\pi f C)}\) If we increase the frequency (f), the impedance of the inductor (\(Z_L\)) will increase, and the impedance of the capacitor (\(Z_C\)) will decrease.
03

Investigate how the current changes

In an AC circuit, the current (I) is related to the voltage (V) and impedance (Z) through Ohm's law, given by \(I = \frac{V}{Z}\). If the frequency of the emf is increased: For the first circuit (inductor): The impedance (\(Z_L\)) increases, which will result in a decrease in current (\(I_L\)). For the second circuit (capacitor): The impedance (\(Z_C\)) decreases, which will result in an increase in current (\(I_C\)).
04

Choose the correct option

Based on our analysis, when the frequency of the emf of AC is increased, the current will decrease in the first circuit (inductor) and increase in the second circuit (capacitor). Therefore, the correct option is: (d) Decrease in the first and increase in the other

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