What happens to the inductive reactance and the current in a purely inductive circuit, if the freque

Q: Why does an inductor oppose AC current?

An inductor resists changes in current due to self-induction. Since AC is constantly changing, the inductor continuously opposes it, creating an effective resistance called reactance.

Q: Why is $X_L$ directly proportional to frequency?

Higher frequency means the current changes more rapidly. A faster change in current induces a larger opposing electromotive force (EMF) according to Faraday's Law, hence greater reactance.

Q: Does the voltage change when we halve the frequency?

No, unless specified, we assume the amplitude of the voltage source remains constant. Only the rate at which it alternates (frequency) is changed.

Q: What would happen in a purely capacitive circuit?

For a capacitor, reactance is inversely proportional to frequency ($X_C = 1 / 2\pi f C$). Halving the frequency would double the reactance, thereby halving the current.

Q: Is inductive reactance measured in Ohms?

Yes! Even though it's not a traditional resistor, reactance still represents opposition to current flow, so its SI unit is the Ohm ($\Omega$).

Question

What happens to the inductive reactance and the current in a purely inductive circuit, if the frequency is halved ?

Options:

(A) Both inductive reactance and current will be halved.

(B) Inductive reactance will be halved and current will be doubled.

(C) Inductive reactance will be doubled and current will be halved.

(D) Both inductive reactance and current will be doubled.

Sigma Percentile · Accepted Answer

### The Setup: A Purely Inductive Circuit

Imagine a simple AC circuit consisting of just an alternating voltage source and an inductor. The AC source provides a voltage $V$ that oscillates at a specific frequency $f$. This oscillating voltage drives an alternating current $I$ through the inductor $L$.

Unlike a simple resistor, an inductor actively opposes any change in the current flowing through it. Because alternating current is constantly changing direction and magnitude, the inductor continuously fights this change. This opposition to the flow of AC is known as **inductive reactance**.

### The Nature of Inductive Reactance

Inductive reactance, denoted by $X_L$, is the AC equivalent of resistance for an inductor. It is mathematically defined by the equation:

$X_L = \omega L = 2\pi f L$

Notice the beautiful direct proportionality here. The inductive reactance $X_L$ is directly proportional to the frequency $f$. This makes perfect physical sense: a higher frequency means the current is changing more rapidly. The faster the current changes, the harder the inductor fights back, resulting in a higher reactance.

### The Effect of Halving Frequency

The problem asks us to consider a scenario where the frequency of the AC source is halved. Let's call our new frequency $f'$.

$f' = \frac{f}{2}$

Because of the direct proportionality we just established, whatever happens to the frequency must also happen to the inductive reactance. If we substitute our new frequency into the reactance formula, we get:

$X_L' = 2\pi \left(\frac{f}{2}\right) L = \frac{X_L}{2}$

So, **halving the frequency exactly halves the inductive reactance**.

### Ohm's Law in AC Circuits

Now, let's figure out what happens to the current. In an AC circuit, Ohm's law takes a slightly modified form. Instead of $I = \frac{V}{R}$, we use the total opposition to current, which in this purely inductive circuit is just $X_L$:

$I = \frac{V}{X_L}$

We need to find the new current, $I'$, when the reactance is halved. Let's substitute our new reactance $X_L'$ into the equation:

$I' = \frac{V}{X_L'} = \frac{V}{\frac{X_L}{2}}$

When you divide by a fraction, it's the same as multiplying by its reciprocal. The $2$ in the denominator flips up to the numerator:

$I' = 2 \left(\frac{V}{X_L}\right) = 2I$

### The Final Verdict

The math reveals a perfect inverse relationship. Because the opposition to the current (the reactance) was cut in half, the current flowing through the circuit was able to flow twice as easily.

Therefore, **the inductive reactance is halved, and the current is doubled.**

Animated Solution for Physics - Alternating Current: What happens to the inductive reactance and the current in a purely inductive circuit, if the frequency is halved ?

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Visualized Solution

$C i r c u i t S e t u p$

$I n d u c t i v e R e a c t an ce$

$H a l v in g t h e F r e q u e n cy$

$C u r r e n t E q u a t i o n$

$N e w C u r r e n t$

$C o n c l u s i o n$

The Sigma Insight: AC Circuits and Power in AC Circuits

The Setup

The Nature of Inductive Reactance

The Effect of Halving Frequency

Ohm's Law in AC Circuits

The Final Verdict

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Animated Solution for Physics - Alternating Current: What happens to the inductive reactance and the current in a purely inductive circuit, if the frequency is halved ?

Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Circuit Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Inductive Reactance

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Halving the Frequency

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Current Equation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } New Current

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conclusion