Problem 39 - Master Resource Book in JEE Main Physics

Question

A circular coil expands radially in a region of magnetic field and no electromotive force is produced in the coil. This can be because \(\quad\) [NCERT Exemplar](a) the magnetic field is constant (b) the magnetic field is in the same plane as the circular coil and it may or may not vary (c) the magnetic field has a perpendicular (to the plane of the coil component whose magnitude is decreasing suitably (d) there is a constant magnetic field in the perpendicular (to the plane of the coill direction

Step-by-Step Solution

Answer

The correct answer is (b): the magnetic field is in the same plane as the circular coil, and it may or may not vary.

Step 1: Understanding the Problem

We have a circular coil expanding in a magnetic field. The problem states that no electromotive force (emf) is induced in the coil, which typically implies no change in magnetic flux through the coil, according to Faraday's Law of Electromagnetic Induction.

Step 2: Applying Faraday's Law

According to Faraday's Law of Electromagnetic Induction, an emf is induced in a circuit when there is a change in magnetic flux through the circuit. The law is mathematically represented as \( \epsilon = -\frac{d\Phi}{dt} \), where \( \Phi \) is the magnetic flux. Since no emf is produced, \( \frac{d\Phi}{dt} = 0 \), meaning the magnetic flux must be constant.

Step 3: Analyzing Magnetic Flux

Magnetic flux (\( \Phi \)) through a surface is given by \( \Phi = B \cdot A \cdot \cos \theta \), where \( B \) is the magnetic field strength, \( A \) is the area of the coil, and \( \theta \) is the angle between the magnetic field and the normal to the area of the coil. Since the coil is expanding, \( A \) is changing. To keep \( \Phi \) constant, either \( B \) or \( \cos \theta \) must adjust to counterbalance this.

Step 4: Evaluating Possible Scenarios

Let's evaluate the given options:- **Option (a)**: If the magnetic field is constant, it does not necessarily mean \( \Phi \) stays constant since \( A \) is changing.- **Option (b)**: If the magnetic field lies in the plane of the coil, \( \theta = 90^\circ \) and \( \cos \theta = 0 \), meaning flux is zero and remains zero even if \( B \) changes.- **Option (c)**: If the perpendicular component of the magnetic field decreases suitably, it can maintain constant flux.- **Option (d)**: A constant perpendicular magnetic field will result in changing flux due to the change in \( A \).

Step 5: Selecting the Correct Answer

Based on the analysis, option (b) satisfies the condition, since with \( \theta = 90^\circ \), the magnetic flux is always zero and unaffected by changes in \( A \) or \( B \).

Key Concepts

Magnetic Flux

Imagine magnetic flux as a measure of how much magnetic field passes through a given surface. It helps us understand the effect of a magnetic field on a coil or loop. The concept of magnetic flux is central to understanding electromagnetic induction. Think of it as the total magnetic field lines passing through a loop or an area.
Magnetic flux is calculated using the formula:

• \( \Phi = B \times A \times \cos \theta \)
• Here, \( B \) is the magnetic field strength or magnetic flux density measured in teslas (T).
• \( A \) is the area through which the field lines pass, in square meters (m²).
• \( \theta \) is the angle between the magnetic field and the perpendicular (normal) to the surface area, measured in degrees.

Let's associate this with our problem. If we have a constant magnetic field and a coil whose area is increasing, changes in flux depend on the relative changes in \( B \) or the angle \( \theta \). If either of these adjusts so that the total value of \( \Phi \) remains constant, no electromotive force will be generated.

Electromotive Force

Electromotive force (emf) might sound intense, but it's simply the energy per unit charge produced by a change in magnetic flux. According to Faraday's Law of Electromagnetic Induction, emf is induced whenever there is a flux change, represented by:

• \( \epsilon = -\frac{d\Phi}{dt} \)
• The negative sign indicates that the induced emf opposes the change in magnetic flux, following Lenz's Law.

If we relate this to the exercise, no emf implies \( \frac{d\Phi}{dt} = 0 \). This zero value means there is no change in magnetic flux. Whether the coil expands or contracts, if the magnetic field's configuration ensures constant flux, no opposing current will be generated across the coil. This is significant in electrical circuits where unwanted emf can influence circuit components and overall functionality.

Circular Coil in Magnetic Field

When you place a circular coil in a magnetic field, several factors influence the interaction. Primarily, the orientation of the coil and its position relative to the field play crucial roles. The problem at hand involves understanding the scenarios when no emf is induced as a coil dynamically changes its area.
Consider a coil laying flat in a magnetic field - meaning the field is in the same plane as the coil. Here, the angle \( \theta \) between the field and the perpendicular to the coil's surface is 90 degrees. This makes \( \cos \theta = 0 \), indicating no flux, since the field is entirely parallel to the plane of the coil. Any changes in the coil's radius (or its area) do not impact the zero flux condition.
Therefore, in scenarios where the magnetic field direction remains in-plane with the coil despite its expansion, the resulting zero flux means no electromotive force is produced. This highlights the importance of field direction and coil orientation in electromagnetic phenomena.