Problem 27 - Master Resource Book in JEE Main Physics

Question

Two coils \(X\) and \(Y\) are placed in a circuit such that a current changes by \(2 \mathrm{~A}\) in coil \(X\) and magnetic flux change of \(0.4\) Wb occurs in \(Y\). The value of mutual inductance of the coils is (a) \(0.8 \mathrm{H}\) (b) \(0.2 \mathrm{~Wb}\) (c) \(0.2 \mathrm{H}\) (d) \(5 \mathrm{H}\)

Step-by-Step Solution

Answer

The mutual inductance is 0.2 H, option (c).

Step 1: Understanding Mutual Inductance

Mutual inductance, denoted as \(M\), is defined as the ratio of the change in magnetic flux \(\Delta \Phi\) through a coil to the change in current \(\Delta I\) through another coil. Mathematically, it can be expressed as \( M = \frac{\Delta \Phi}{\Delta I} \). In this situation, \(\Delta \Phi = 0.4\, \text{Wb}\) and \(\Delta I = 2\, \text{A}\).

Step 2: Substitute Values into the Formula

Substitute the values of \(\Delta \Phi\) and \(\Delta I\) into the mutual inductance formula to find \(M\): \[ M = \frac{0.4}{2} \].

Step 3: Calculate the Mutual Inductance

Perform the division to calculate the mutual inductance: \[ M = \frac{0.4}{2} = 0.2 \].

Step 4: Identify the Correct Answer from the Options

Compare the calculated mutual inductance \(0.2\, \text{H}\) with the given options. The correct option is (c) \(0.2\, \text{H}\).

Key Concepts

Magnetic Flux

Magnetic flux, represented by \( \Phi \), refers to the measure of the magnetic field passing through a certain area. Imagine it as invisible lines of force that travel from one magnet or coil to another. You can think of these lines as the wind waves gently brushing over and surrounding objects. This flux is crucial in understanding how magnetic fields interact with electrical circuits.

• Unit: Magnetic flux is measured in Webers (Wb).
• Formula: Flux is typically determined by the equation \( \Phi = B \times A \times \cos(\theta) \), where \(B\) is the magnetic field strength, \(A\) is the area, and \(\theta\) is the angle between the field and the area's normal line.

The exercise given involves a change in magnetic flux of \(0.4\) Wb. This change across one coil due to a current change in another coil is key to understanding mutual inductance.

Current Change

A change in current, \( \Delta I \), is a crucial aspect in the study of electromagnetism and inductance. It refers to the variation or alteration in the electrical current flowing through a circuit over time.When the current in one coil changes, it affects the magnetic field around it. This, in turn, influences nearby coils, an effect central to the principle of mutual inductance.

• Impact: Altered current can cause changes in magnetic flux around the coil.
• Relation with Inductance: The mutual inductance helps understand how much the current change in one coil can influence the magnetic field in another.

In the provided problem, the current change is given as \(2\) A, which causes the magnetic flux in the second coil to change. This relationship is evaluated using the mutual inductance formula.

Inductance Calculation

Inductance is the property of a coil that quantifies its ability to induce a voltage in response to a change in current. There are two main types: self-inductance and mutual inductance. Our focus here is on mutual inductance.To calculate mutual inductance \(M\) for the two coils given in the problem, you use the formula:\[M = \frac{\Delta \Phi}{\Delta I}\]Here:- \(\Delta \Phi\) signifies the change in the magnetic flux, which is \(0.4\) Wb.- \(\Delta I\) represents the change in current, \(2\) A.To find the value of \(M\), substitute these values into the formula:\[ M = \frac{0.4}{2} = 0.2 \]So, the mutual inductance, \(M\), equals \(0.2\, \text{H}\). This value specifies the effectiveness of the coil's ability to induce electromagnetic force in each other when current changes occur.