Problem 17 - Master Resource Book in JEE Main Physics

Question

A circular ring of diameter \(20 \mathrm{~cm}\) has a resistance of \(0.01 \Omega\). The charge that will flow through the ring if it is turned from a position perpendicular to a uniform magnetic field of \(2.0 \mathrm{~T}\) to a position parallel to the field is about (a) \(63 \mathrm{C}\) (b) \(0.63 \mathrm{C}\) [c) \(6.3 \mathrm{C}\) (d) \(0.063 \mathrm{C}\)

Step-by-Step Solution

Answer

The charge that will flow through the ring is approximately 6.3 C (option c).

Step 1: Calculate the area of the ring

The area of a circle is given by the formula \( A = \pi r^2 \). Since the diameter of the ring is 20 cm, its radius \( r \) is \( \frac{20}{2} = 10 \) cm or \( 0.1 \) m. Thus, the area \( A \) is \( \pi \times (0.1)^2 = 0.01\pi \) square meters.

Step 2: Calculate the magnetic flux initially and finally

The magnetic flux \( \Phi \) through an area \( A \) in a magnetic field \( B \) is given by \( \Phi = B \cdot A \cdot \cos \theta \). Initially, the ring is perpendicular to the field, \( \theta = 90^\circ \), so the initial flux \( \Phi_i = B \times A \times \cos 90^\circ = 0 \). Finally, the ring is parallel to the field, \( \theta = 0^\circ \), so the final flux \( \Phi_f = B \times A \times \cos 0^\circ = 2 \pi \times 0.01 = 0.02 \pi \).

Step 3: Calculate the change in magnetic flux

The change in magnetic flux \( \Delta \Phi \) is the difference between the final and initial flux. Thus, \( \Delta \Phi = \Phi_f - \Phi_i = 0.02 \pi - 0 = 0.02 \pi \).

Step 4: Use Faraday's law to find the induced EMF

According to Faraday's law, the induced electromotive force (EMF) \( \mathcal{E} \) is equal to the rate of change of magnetic flux, \( \mathcal{E} = - \frac{d\Phi}{dt} \). Since there are no time details provided, we consider the total change \( \mathcal{E} = \Delta \Phi \). So, \( \mathcal{E} = 0.02 \pi \).

Step 5: Calculate the charge using Ohm's Law

Ohm's law relates the EMF to the current and resistance by \( \mathcal{E} = I \times R \). The charge \( q \) flowing is related by \( q = I \times t \), and since \( \mathcal{E} = q \times \frac{R}{t} \), we simplify it to \( q = \frac{\mathcal{E}}{R} \). Thus, substituting the EMF and resistance values, \( q = \frac{0.02 \pi}{0.01} = 2 \pi = 6.28 \).

Step 6: Compare the answer with given options

The calculated charge \( q = 6.28 \) is approximately equal to the option (c) \( 6.3 \mathrm{C} \).

Key Concepts

Magnetic Flux

Magnetic flux is a crucial concept in electromagnetic induction. It represents the measure of the amount of magnetic field passing through a given area. Imagine a surface exposed to a magnetic field; the number of magnetic field lines passing through it constitutes the magnetic flux. Mathematically, magnetic flux \( \Phi \) is given by the formula \( \Phi = B \cdot A \cdot \cos \theta \), where:

• \( B \) is the magnetic field strength,
• \( A \) is the area of the surface,
• \( \theta \) is the angle between the field lines and the normal (perpendicular) to the surface.

When \( \theta = 0^\circ \), the surface is parallel to the field lines, resulting in maximum magnetic flux, since \( \cos 0^\circ = 1 \). Conversely, at \( \theta = 90^\circ \), the surface is perpendicular to the field, and magnetic flux is zero, because \( \cos 90^\circ = 0 \). Changes in magnetic flux with time are foundational to understanding electromagnetic induction.

Faraday's Law

Faraday's Law is a cornerstone of electromagnetic induction. It explains how changing magnetic flux induces an electromotive force (EMF) or voltage in a conductor. This law is especially crucial in understanding the behavior of electrical circuits exposed to varying magnetic fields. The law is mathematically expressed as \( \mathcal{E} = - \frac{d\Phi}{dt} \), where:

• \( \mathcal{E} \) is the induced EMF,
• \( \Phi \) is the magnetic flux,
• \( \frac{d\Phi}{dt} \) represents the rate of change of magnetic flux.

The negative sign is indicative of Lenz's Law. It denotes that the induced EMF will generate a current whose magnetic field opposes the initial change in flux. This fundamental law not only predicts the generation of EMF but also explains phenomena such as the functioning of electric generators and transformers.

Ohm's Law

Ohm's Law is a simple yet vital law in understanding electrical circuits. It describes the relationship between voltage, current, and resistance. Expressed mathematically, Ohm's Law is \( V = I \cdot R \), where:

• \( V \) is the voltage or EMF in volts,
• \( I \) is the current in amperes,
• \( R \) is the resistance in ohms.

This law implies that the current \( I \) flowing through a conductor is directly proportional to the voltage \( V \) applied across it, provided the resistance \( R \) remains constant. In the context of electromagnetic induction, Ohm's Law is used to relate the induced EMF to the resulting current flow in a conductor with a given resistance. By rearranging this equation, we can also find other parameters. For instance, knowing \( \mathcal{E} \) and \( R \), we can calculate the charge \( q \) flowing through a circuit using the formula \( q = \frac{\mathcal{E}}{R} \), crucial for understanding induced currents in circuits.