Problem 26 - 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 \(0.2 \, \mathrm{H}\), option (c).

Step 1: Identify Given Values

We are given that the current change in coil \(X\) is \(2 \, \mathrm{A}\), and the change in magnetic flux in coil \(Y\) is \(0.4 \, \mathrm{Wb}\).

Step 2: Write Formula for Mutual Inductance

The formula for mutual inductance \(M\) is given by \(M = \frac{\Delta \Phi}{\Delta I}\), where \(\Delta \Phi\) is the change in magnetic flux and \(\Delta I\) is the change in current.

Step 3: Substitute Values Into Formula

Using the values \(\Delta \Phi = 0.4 \, \mathrm{Wb}\) and \(\Delta I = 2 \, \mathrm{A}\), substitute them into the formula: \[ M = \frac{0.4}{2} \].

Step 4: Calculate Mutual Inductance

Calculate the value of \(M\): \( M = 0.2 \, \mathrm{H} \).

Step 5: Select Correct Answer

The correct value of mutual inductance \(M\) is \(0.2 \, \mathrm{H}\), which corresponds to option (c).

Key Concepts

Magnetic Flux

Magnetic flux is a crucial concept in understanding mutual inductance. It refers to the measure of the quantity of magnetism, taking into account the strength and the extent of a magnetic field. You can think of it as the 'flow' of magnetism through a given area. The unit of magnetic flux is the Weber (Wb).

• Magnetic flux is directly related to the magnetic field strength and the area it permeates.
• It can change due to variations in the magnetic field or the position and orientation of the area compared to the field.
• The foundational formula is: \[ \Phi = B imes A \]where \( \Phi \) is the magnetic flux, \( B \) is the magnetic field strength, and \( A \) is the area.

In our problem, the magnetic flux change in coil \(Y\) is given as \(0.4 \, \text{Wb}\). This change directly affects the mutual inductance when paired with the current change in the other coil. Mutual inductance is a measure of how well one coil can induce a voltage in another through magnetic flux.

Current Change

When discussing mutual inductance, current change is essential. It's the change in electric current that affects the magnetic field and, consequently, the magnetic flux.

• A change in current in one coil will cause a change in magnetic flux in the accompanying coil.
• This is a principle part of electromagnetic induction which is the underlying principle of mutual inductance.
• The relationship is described by the formula \[M = \frac{\Delta \Phi}{\Delta I}\]where \( M \) is mutual inductance, \( \Delta \Phi \) is the change in magnetic flux, and \( \Delta I \) is the change in current.

In our exercise, the change in current for coil \(X\) is \(2 \, \text{A}\). This change is crucial because the interaction between this current alteration and the resultant magnetic flux change in coil \(Y\) helps determine the mutual inductance.

Physics Problem Solving

Solving physics problems requires a systematic approach. Breaking down the problem into manageable parts helps simplify complex concepts. Here’s a methodical way to approach the given exercise about mutual inductance.Start by recognizing what is provided to you: the change in current \( \Delta I = 2 \, \text{A} \), and the change in magnetic flux \( \Delta \Phi = 0.4 \, \text{Wb} \). It’s important to clearly articulate these to reduce confusion.

• Use the correct formula: Remember that the formula for mutual inductance \( M \) is \[M = \frac{\Delta \Phi}{\Delta I}\]Precision in using formulas is key in physics problem solving.
• Substitute the known values: Accurately plug the numbers into the formula. In this case, \[M = \frac{0.4}{2} = 0.2 \, \text{H}\]This confirms the mutual inductance.
• Check your work: Validate the calculations and ensure you choose the correct answer based on the options provided.

Thoughtful, deliberate steps help demystify physics problems and lead to correct solutions. This methodical breakdown enables a deeper understanding of the underlying principles, such as mutual inductance.