Problem 36 - Master Resource Book in JEE Main Physics

Question

Two circuits have mutual inductance of \(0.09 \mathrm{H}\). Average emf induced in the secondary by a change of current from 0 to \(20 \mathrm{~A}\) in \(0.006 \mathrm{~s}\) in primary will be (a) \(120 \mathrm{~V}\) (b) \(200 \mathrm{~V}\) (c) \(180 \mathrm{~V}\) (d) \(300 \mathrm{~V}\)

Step-by-Step Solution

Answer

The average induced emf is \( 300 \, \mathrm{V} \), so the answer is (d).

Step 1: Identify Given Values

Extract the given information from the problem. The mutual inductance \( M \) is \( 0.09 \, \mathrm{H} \), the change in current \( \Delta I \) is from \( 0 \, \mathrm{A} \) to \( 20 \, \mathrm{A} \), and the change in time \( \Delta t \) is \( 0.006 \, \mathrm{s} \).

Step 2: Recall the Formula for Induced EMF

The formula for the average electromotive force (emf) induced in the secondary coil due to a change in current through the primary coil is given by \( \mathcal{E} = M \frac{\Delta I}{\Delta t} \).

Step 3: Calculate the Change in Current

Calculate the change in current \( \Delta I \) which is the final current minus the initial current. Since it changes from \( 0 \) to \( 20 \), \( \Delta I = 20 \, \mathrm{A} - 0 \, \mathrm{A} = 20 \, \mathrm{A} \).

Step 4: Substitute Values into the Formula

Substitute the given values into the emf formula. We have \[ \mathcal{E} = 0.09 \, \mathrm{H} \times \frac{20 \, \mathrm{A}}{0.006 \, \mathrm{s}} \].

Step 5: Calculate the Induced EMF

Calculate the emf using the substituted values: \[ \mathcal{E} = 0.09 \times \frac{20}{0.006} = 0.09 \times 3333.33 \approx 300 \, \mathrm{V} \].

Step 6: Choose the Correct Answer

Match the calculated result of \( 300 \, \mathrm{V} \) with the options given. The correct option is (d) \( 300 \, \mathrm{V} \).

Key Concepts

Electromotive Force

Electromotive force, often abbreviated as emf, is a fundamental concept in electromagnetism. It represents the energy provided by a source to move electrical charge through a circuit.

Emf can be thought of as the electrical "pressure" that pushes charges through a conductor. It is measured in volts (V) and is similar to how water pressure pushes water through pipes.

In a scenario involving mutual inductance, the emf can be induced in a secondary coil when the current through a primary coil changes. This happens due to the magnetic field created by the current flow.

• The faster the change in current, the greater the induced emf.
• The induced emf is directly proportional to the rate of current change and the mutual inductance between the coils.

Emf is crucial because it essentially determines how much potential difference can drive a current in a circuit, thus enabling electrical devices to function.

Change in Current

The change in current, often denoted as \( \Delta I \), is a measure of how much the current flowing through a circuit changes over a specific period.

This concept is pivotal in understanding how and why the electromotive force is induced in circuits that are mutually inductive.

In the context of the exercise, the current changes from 0 to 20 A, making \( \Delta I = 20 \text{ A} \).

• \( \Delta I \) indicates the total amount of change in current, which tells us the speed and magnitude of this change.
• The change in current affects how quickly the magnetic field around the primary circuit is altered.

The rapid change is particularly impactful in electromagnetic applications, such as transformers, where it's essential to precisely control the rate of current change to optimize performance.

Mutual Inductance Formula

The mutual inductance formula expresses the relationship between the induced emf in one coil and the change in current through another coil. It is given by the equation:\[ \mathcal{E} = M \frac{\Delta I}{\Delta t}\] where:

• \( \mathcal{E} \) is the induced electromotive force (emf) in volts.
• \( M \) is the mutual inductance in henries (H).
• \( \Delta I \) is the change in current in amperes (A).
• \( \Delta t \) is the time period over which the change occurs, in seconds (s).

The formula highlights how mutual inductance \( M \) influences the amount of induced emf when there is a change in current in the primary coil.

In our exercise, using the given values of \( M = 0.09 \text{ H} \), \( \Delta I = 20 \text{ A} \), and \( \Delta t = 0.006 \text{ s} \), the induced emf turns out to be approximately 300 V.

This relationship is fundamental in designing and studying systems involving inductive components, like transformers and inductors, ensuring efficient energy transfer through varying magnetic fields.