Problem 30 - Master Resource Book in JEE Main Physics

Question

When current in a coil changes from \(2 \mathrm{~A}\) to \(-2 \mathrm{~A}\) in \(0.05 \mathrm{~s}\), an emf of \(8 \mathrm{~V}\) is induced in the coil. The coefficient of self-inductance of the coil is (a) \(0.1 \mathrm{H}\) (b) \(0.2 \mathrm{H}\) (c) \(0.4 \mathrm{H}\) (d) \(0.8 \mathrm{H}\)

Step-by-Step Solution

Answer

(a) \(0.1 \mathrm{~H}\)

Step 1: Understand the Given Data

We are given that the current changes from \(2 \mathrm{~A}\) to \(-2 \mathrm{~A}\) in time \(0.05 \mathrm{~s}\), and an electromotive force (emf) of \(8 \mathrm{~V}\) is induced. We are required to find the coefficient of self-inductance \(L\) of the coil.

Step 2: Calculate the Change in Current

The change in current, \( \Delta I \), is given by the final current minus the initial current. Hence, \( \Delta I = -2 \mathrm{~A} - 2 \mathrm{~A} = -4 \mathrm{~A} \).

Step 3: Use Faraday's Law of Induction

According to Faraday's Law, the induced emf \( \mathcal{E} \) in a coil is given by the formula: \( \mathcal{E} = -L \frac{\Delta I}{\Delta t} \), where \( L \) is the self-inductance, \( \Delta I \) is the change in current, and \( \Delta t \) is the change in time.

Step 4: Substitute the Known Values

Substitute the known values into the equation: \( 8 = -L \frac{-4}{0.05} \).

Step 5: Solve for Self-Inductance \(L\)

Simplify the equation from the previous step: \[ 8 = L \times \frac{4}{0.05} \]Rearrange to solve for \(L\): \[ L = \frac{8 \times 0.05}{4} \]

Step 6: Calculate \(L\)

Calculate the value of \(L\):\[ L = \frac{0.4}{4} = 0.1 \mathrm{~H} \]

Step 7: Compare with Given Options

The calculated self-inductance \(L\) is \(0.1 \mathrm{~H}\), which corresponds to option (a).

Key Concepts

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle in electromagnetism. This law explains how changing magnetic fields can produce an electromotive force (emf) in a coil. It's the backbone of technologies like transformers and electric generators. In simple terms, when there's a change in the magnetic environment of a coil, an emf is induced. This occurs because the magnetic field lines are either entering or exiting the coil, creating an electric field inside it.

Faraday's Law can be expressed with the formula:

• \( \mathcal{E} = -L \frac{\Delta I}{\Delta t} \)

Here, \(\mathcal{E}\) represents the induced emf, \(L\) is the self-inductance, \(\Delta I\) is the change in current, and \(\Delta t\) is the time interval over which the current changes.

This law emphasizes how the induced emf is not just about the magnitude of the change but also the rate of change over time. Faster changes result in a greater emf. The negative sign represents Lenz's Law, indicating that the induced emf acts in a direction to oppose the change causing it.

Change in Current

The concept of a change in current is central to understanding induced electromotive force. When we talk about a change in current, we're referring to the difference in the current flowing through a conductor over a period of time. In the given exercise, the current changes from \(2 \mathrm{A}\) to \(-2 \mathrm{A}\) within \(0.05 \mathrm{s}\). This change can be calculated as follows:

• \( \Delta I = I_{final} - I_{initial} \)

Substituting the given values, we have \( \Delta I = -2 \mathrm{~A} - 2 \mathrm{~A} = -4 \mathrm{~A} \).

The negative sign here shows that the direction of the current has reversed. This is an important aspect to consider because the direction of current affects the polarity of the induced emf. Knowing \(\Delta I\) is crucial as it directly influences the magnitude of the induced emf, according to Faraday's Law. Understanding how the current changes provides insight into how the coil responds electrically.

Induced Electromotive Force

Induced electromotive force (emf) is a core concept in electromagnetism which describes the voltage generated in a coil when the magnetic environment around it changes. This change can be due to a variation in the magnetic field strength, the area of the coil, or the orientation of the coil.

The formula for calculating the induced emf is given by Faraday's Law:

• \( \mathcal{E} = -L \frac{\Delta I}{\Delta t} \)

In this case, the induced emf \( \mathcal{E} \) is directly related to the rate of change of current \( \frac{\Delta I}{\Delta t} \) through the coil, and the self-inductance \( L \) of the coil.

For example, in the exercise, an induced emf of \(8 \mathrm{~V}\) is observed when the current changes from \(2 \mathrm{~A}\) to \(-2 \mathrm{~A}\) in \(0.05 \mathrm{~s}\). This illustrates how dynamic conditions like changing currents result in generating a voltage (emf) across the coil. This induced emf has practical implications in devices like motors and inductors, where controlling the generated voltage is essential for device operation.