Problem 21 - Master Resource Book in JEE Main Physics

Question

A square loop of wire of side \(5 \mathrm{~cm}\) is lying on a horizontal table. An electromagnet above and to one side of the loop is turned on, causing a uniform magnetic field downwards at an angle of \(60^{\circ}\) to the vertical as shown in figure. The magnetic induction is \(0.50 \mathrm{~T}\). The average induced emf in the loop, if the field increases from zero to its final value in \(0.2 \mathrm{~s}\) is (a) \(5.4 \times 10^{-3} \mathrm{~V}\) (b) \(312 \times 10^{-3} \mathrm{~V}\) (c) 0 (d) \(0.25 \times 10^{-3} \mathrm{~V}\)

Step-by-Step Solution

Answer

The average induced emf is \(3.125 \times 10^{-3} \mathrm{~V}\).

Step 1: Understand the Problem

We have a square loop of wire and need to find the average induced electromotive force (emf) when the magnetic field changes. The loop's side is \(5 \mathrm{~cm}\), the magnetic field increases to \(0.50 \mathrm{~T}\) in \(0.2 \mathrm{~s}\), and is at an angle of \(60^{\circ}\) to the vertical.

Step 2: Determine Relevant Equations

The average induced emf \( \mathcal{E} \) can be calculated using Faraday's law of electromagnetic induction: \( \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \). The magnetic flux \( \Phi \) through the square loop at an angle is given by \( \Phi = B \cdot A \cdot \cos \theta \). Here, \( B \) is the magnetic field, \( A \) is the area of the loop, and \( \theta \) is the angle between the magnetic field and the normal to the loop's surface.

Step 3: Calculate the Area of the Loop

The side of the square loop is \(5 \mathrm{~cm}\), so the area \( A \) is calculated as \( A = (5 \times 10^{-2} \mathrm{~m})^2 = 25 \times 10^{-4} \mathrm{~m}^2 \).

Step 4: Calculate Change in Magnetic Flux

The initial magnetic flux \( \Phi_i = 0 \) because the initial magnetic field is zero. The final magnetic flux \( \Phi_f = B \cdot A \cdot \cos \theta = 0.50 \mathrm{~T} \cdot 25 \times 10^{-4} \mathrm{~m}^2 \cdot \cos 60^{\circ} = 0.50 \cdot 25 \times 10^{-4} \cdot 0.5 = 6.25 \times 10^{-4} \mathrm{~Wb} \).

Step 5: Calculate Initial and Final Flux Difference

The change in magnetic flux \( \Delta \Phi \) is the difference between final and initial flux: \( \Delta \Phi = \Phi_f - \Phi_i = 6.25 \times 10^{-4} - 0 = 6.25 \times 10^{-4} \mathrm{~Wb} \).

Step 6: Calculate Average Induced EMF

Using Faraday's law, the average induced emf is \( \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} = -\frac{6.25 \times 10^{-4} \mathrm{~Wb}}{0.2 \mathrm{~s}} = -3.125 \times 10^{-3} \mathrm{~V} \). The negative sign indicates direction, so the magnitude is \(3.125 \times 10^{-3} \mathrm{~V} \). Given options, it should round closer to \(0.25 \times 10^{-3} \mathrm{~V}\).

Key Concepts

Faraday's Law

Faraday's Law is a fundamental principle of electromagnetic induction, discovered by Michael Faraday. It explains how an electric current is induced in a conductor by a changing magnetic field. This law forms the basis of many electrical devices such as transformers and generators.

Faraday's Law can be expressed by the formula:

• \( \mathcal{E} = -\frac{d\Phi}{dt} \)

where:

• \( \mathcal{E} \) is the induced electromotive force (emf)
• \( \Phi \) is the magnetic flux
• \( \frac{d\Phi}{dt} \) is the rate of change of the magnetic flux over time

The negative sign in Faraday's equation is a result of Lenz's Law, which states that the induced current will flow in a direction such that it opposes the change in magnetic flux that produced it. This ensures conservation of energy.

In the given exercise, Faraday's law helps calculate the average induced emf when the magnetic field in a square loop changes.

Magnetic Flux

Magnetic Flux is a measure of the quantity of magnetism, considering the strength and extent of a magnetic field. It is represented by the Greek letter \( \Phi \). This concept is crucial in understanding electromagnetic induction.

The formula for magnetic flux is:

• \( \Phi = B \cdot A \cdot \cos \theta \)

where:

• \( B \) is the magnetic field strength in teslas (T)
• \( A \) is the area through which the magnetic field lines pass
• \( \theta \) is the angle between the magnetic field and the perpendicular to the surface

In the exercise, the initial magnetic flux is zero, because the magnetic field starts from zero. When it increases at an angle \( \theta = 60^{\circ} \), the change in magnetic flux can be calculated using the given area and final magnetic field strength.

Magnetic flux provides a way to visualize how much magnetic field passes through a loop, making it easier to anticipate the induced emf.

Induced EMF

Induced Electromotive Force (emf) is the voltage generated in a conductor due to a change in magnetic flux. This is a central aspect of electromagnetic induction, as described by Faraday's Law.

In practical terms, the induced emf is crucial for the operation of electric circuits and various devices such as motors, where it can drive a current through the circuit.

According to Faraday's law, the magnitude of the induced emf can be calculated as:

• \( \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \)

In the given exercise, as the magnetic field strength changes from zero to a certain final value, the induced emf is computed by considering the rate of change of the magnetic flux (from zero to its final value) over a time interval of \( 0.2 \) seconds.

Understanding induced emf helps explain the transformation of energy within electromagnetic systems and provides insight into how changing magnetic fields impact electric currents.