Question

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

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

Step-by-step solution

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.

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.

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 (\(I_C\)).

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

Key concepts

Impedance

Impedance is a fundamental concept in AC circuit analysis. It represents the opposition that any component, such as a resistor, capacitor, or inductor, offers to the flow of alternating current.

• Unlike resistance, which affects both direct current (DC) and alternating current (AC), impedance is specifically used in the context of AC circuits.
• Impedance combines both resistance (real part) and reactance (imaginary part) and is expressed as a complex number.

The formula for impedance (Z) depends on the component type and its configuration in the circuit. Important to note is that the total impedance impacts how much current flows through the circuit for a given voltage according to Ohm's Law in AC systems: \[ I = \frac{V}{Z} \] where \(I\) is the current, \(V\) is the voltage, and \(Z\) is the impedance.

Inductor

In an AC circuit, an inductor is a component that opposes changes in current. It does this by inducing a voltage in itself as the current through it changes. This property is known as inductance and is measured in henrys (H).

• An inductor’s impedance is given by the formula: \(Z_L = j\omega L\), where \(j\) represents the imaginary unit, \(\omega\) is the angular frequency, and \(L\) is the inductance.
• As a coil of wire, an inductor stores energy in a magnetic field when current flows through it, which affects how current can change in an AC circuit.

When you increase the frequency of an alternating current circuit, the reactance part of the impedance of an inductor increases because \(Z_L = j(2\pi f L)\). Thus, as frequency increases, the inductor’s opposition to current also increases.

Capacitor

A capacitor in an AC circuit is a component that stores energy in an electric field between a pair of conductors. Capacitance is a measure of a capacitor's ability to store charge and is measured in farads (F).

• Impedance of a capacitor is expressed as \(Z_C = \frac{1}{j\omega C}\), where \(C\) is the capacitance and \(\omega\) is the angular frequency.
• This formula indicates that as the frequency increases, the reactance of the capacitor (opposition to current flow) decreases.

Hence, with increased frequency, a capacitor allows more current to pass through it, due to the reduction in its impedance.

Frequency Effect on Current

Frequency affects how both inductors and capacitors control current flow in AC circuits.

• In an inductor, increasing frequency results in an increase in impedance. Consequently, Ohm's Law \(I = \frac{V}{Z}\) indicates that the current decreases because the opposition (impedance) is greater.
• Conversely, for a capacitor, increasing frequency decreases impedance, resulting in an increase in current since the opposition to current flow reduces.

This is why the behavior of current changes in circuits containing these elements when the frequency of the AC source is altered. Understanding this effect is crucial for designing and analyzing AC circuits in electronic and electrical systems.