Problem 6 - Master Resource Book in JEE Main Physics

Question

A pair of adjacent coils has a mutual inductance of \(1.5 \mathrm{H}\). If the current in one coil changes from 0 to \(20 \mathrm{~A}\) in \(0.5 \mathrm{~s}\), what is the change of flux linkage with the other coil? (a) \(30 \mathrm{~Wb}\) (b) \(33 \mathrm{~Wb}\) (c) \(23 \mathrm{~Wb}\) (d) \(42 \mathrm{~Wb}\)

Step-by-Step Solution

Answer

The change of flux linkage with the other coil is \(30 \mathrm{~Wb}\).

Step 1: Understand the Variables

We have a mutual inductance \(M = 1.5 \mathrm{H}\), and the change in current in one coil \(\Delta I = 20 \mathrm{~A} - 0 \mathrm{~A} = 20 \mathrm{~A}\). The time over which this change occurs is \(\Delta t = 0.5 \mathrm{~s}\). We need to find the change of flux linkage \(\Delta \Phi\) with the other coil.

Step 2: Use the Formula for Flux Linkage

The change in flux linkage \(\Delta \Phi\) is given by the formula: \[ \Delta \Phi = M \cdot \Delta I \]Where \(M\) is the mutual inductance and \(\Delta I\) is the change in current.

Step 3: Substitute Values into the Formula

Substitute the given values into the formula:\[ \Delta \Phi = 1.5 \mathrm{H} \times 20 \mathrm{~A} = 30 \mathrm{~Wb} \]

Step 4: Identify the Correct Option

Compare the calculated change in flux linkage \(30 \mathrm{~Wb}\) to the options provided. The answer matches option (a).

Key Concepts

Flux Linkage

Flux linkage is a crucial concept in understanding electromagnetic phenomena, especially in coil systems. It refers to the total magnetic flux that links a coil, and it is a measure that combines the effect of several factors like magnetic field strength and the number of turns in a coil. This concept helps to describe how a change in magnetic fields can influence electric circuits. In the context of inductance and coils, the flux linkage is proportional to the product of the magnetic flux through a loop and the number of turns of the coil.

For the scenario of mutual inductance, where two coils are involved, the flux linkage can change when there is a variation in current in either coil. Here, the flux linkage behaves as a bridge, transmitting magnetic effects from one coil to another. In simpler terms, if a current in one coil changes, it alters the magnetic environment of the adjacent coil, thereby changing the flux linkage.

Inductance Formula

The inductance formula forms the backbone of calculations involving mutual inductance. Inductance is a property of a coil that measures its ability to induce voltage when the current through it is changing. The mutual inductance, denoted as \( M \), quantifies the induced voltage in one coil due to the current change in another coil.

To calculate the change in flux linkage, we use the formula:

\[ \Delta \Phi = M \times \Delta I \]
Here:

• \( \Delta \Phi \) is the change in flux linkage,
• \( \Delta I \) is the change in current,
• \( M \) is the mutual inductance.

The units of measurement are Henry (H) for inductance and Amperes (A) for current, resulting in Weber (Wb) for the flux linkage, which is a direct application of these core physical principles.

Current Change

A fundamental aspect of mutual inductance is the change of current in one of the coils. The term "current change" refers to the variation in electrical current over a period of time. This change is crucial because it is the driving factor for the corresponding change in magnetic flux, and subsequently, the change in flux linkage in the other coil.

In our example, the current changes from 0 to 20 A over a span of 0.5 seconds. The degree of current change can significantly affect the induced electromotive force and the flux linkage. The rate at which this change occurs—often denominated as \( \frac{\Delta I}{\Delta t} \)—plays a pivotal role in how strongly the coils interact magnetically. This interaction, governed by Faraday's laws, is the essence of transforming electric and magnetic energy in such systems.

Electromagnetic Induction

Electromagnetic induction is the core principle that underlies the phenomena of mutual inductance. This process involves generating an electromotive force (EMF) or voltage across an electrical conductor in a changing magnetic field. It occurs not only in generators but also in scenarios of mutual inductance, where two or more coils influence each other.

This fundamental principle was discovered by Michael Faraday, and it explains how a time-varying current in one coil produces a magnetic field that induces a voltage in an adjacent coil. The induced EMF is proportional to the rate of change of the magnetic flux linkage. Understanding this principle gives insight into many electrical engineering applications, where the effective transfer of energy between components is required.

Electromagnetic induction is at the heart of how transformers, inductors, and generators function, and it continues to be a pivotal concept in both theoretical and applied physics.