Problem 40 - Master Resource Book in JEE Main Physics

Question

A current of \(10 \mathrm{~A}\) in the primary coil of a circuit is reduced to zero. If the coefficient of mutual inductance is \(3 \mathrm{H}\) and emf induced in secondary coil is \(30 \mathrm{kV}\), time taken for the change of current is (a) \(10^{3} \mathrm{~s}\) (b) \(10^{2} \mathrm{~s}\) (c) \(10^{-3} \mathrm{~s}\) (d) \(10^{-2} \mathrm{~s}\)

Step-by-Step Solution

Answer

The time taken for the change of current is (c) \(10^{-3} \mathrm{~s}\).

Step 1: Understanding Mutual Inductance Concept

The concept of mutual inductance tells us that when there's a change in current in one coil, it can induce an electromotive force (emf) in another nearby coil. This relationship is quantified by the coefficient of mutual inductance (M). It is expressed in the equation \( \varepsilon = -M \frac{\Delta I}{\Delta t} \), where \( \varepsilon \) is the induced emf, \( M \) is the mutual inductance, \( \Delta I \) is the change in current, and \( \Delta t \) is the change in time.

Step 2: Identifying Given Values

We are provided with the following values: the initial current (\( I = 10 \ A \)), final current (\( I = 0 \ A \)), mutual inductance (\( M = 3 \ H \)), and the induced emf (\( \varepsilon = 30 \ kV \)). We need \( \Delta t \) to understand the time taken for the change in current.

Step 3: Applying the Formula for Induced EMF

We use the relation \( \varepsilon = -M \frac{\Delta I}{\Delta t} \). Substitute the given values where \( \varepsilon = 30 \times 10^3 \ V \), \( M = 3 \ H \), and \( \Delta I = 10 - 0 \ A = 10 \ A \), to solve for \( \Delta t \).

Step 4: Rearranging the Equation and Solving

Rearrange the equation to find time, \( \Delta t \): \[\Delta t = \frac{M \cdot \Delta I}{\varepsilon}\]Substitute the known values: \[\Delta t = \frac{3 \ times 10}{30 \times 10^3 }\]which simplifies to \( \Delta t = 10^{-3} \ s \).

Step 5: Choosing the Correct Option

With the computed \( \Delta t = 10^{-3} \ s \), refer to the options provided. The correct answer corresponds to option (c) \( 10^{-3} \ s \).

Key Concepts

Induced EMF

Induced electromotive force (EMF) is a critical concept in understanding how electric circuits interact when placed near each other. The induced EMF is generated when there is a change in magnetic flux through a coil due to a changing current in a nearby coil. This process stems from Faraday's Law of Electromagnetic Induction, which states that a changing magnetic field can produce an EMF.

The formula to calculate induced EMF in mutual inductance is given by \( \varepsilon = -M \frac{\Delta I}{\Delta t} \), where:

• \( \varepsilon \) is the induced EMF.
• \( M \) represents the mutual inductance.
• \( \Delta I \) is the change in current.
• \( \Delta t \) is the change in time.

This equation highlights that the induced EMF is also dependent on how quickly the current changes, showing a direct relationship with the speed of current change. The negative sign indicates the direction of the induced EMF opposes the change in current, which is a demonstration of Lenz's Law.

Current Change

The change in current is a fundamental part of the process that leads to the induction of EMF in a nearby coil. When there is a rapid change in the current flowing through the primary coil, it affects the magnetic field surrounding it, subsequently influencing nearby coils.

For example, consider a scenario where the current in a coil suddenly reduces from 10 A to 0 A. This significant alteration in current generates a varying magnetic field. The specifics of this change—such as how great the drop or rise in current is—will determine the magnitude of the induced EMF when mutual inductance comes into play.

Calculating the change in current involves simply taking the difference between the initial and final current values, represented by \( \Delta I \). In our example, the change \( \Delta I = 10 \ A - 0 \ A = 10 \ A \). Understanding and correctly calculating this value is crucial for solving problems related to electromagnetic induction.

Electromagnetic Induction

Electromagnetic induction is the process by which a change in magnetic environment of a coil induces an EMF in that coil or another nearby coil. This concept is pivotal in many technological applications, such as in the generation of electricity in power plants and the functioning of transformers.

The mechanism of electromagnetic induction is beautifully illustrated by experiments showing how a moving magnet through a coil generates an electrical current. This principle is what allows power utilities to convert mechanical energy into electrical energy efficiently.

Two crucial laws derive the foundation of electromagnetic induction:

• Faraday's Law, which quantifies the induced EMF based on the rate of change of magnetic flux.
• Lenz's Law, which predicts the direction of the induced current, emphasizing the principle of energy conservation by opposing the change causing it.

Electromagnetic induction allows devices to transfer energy without direct physical contact, underlining the power of changing magnetic fields in electrical technology and devices.