Problem 12 - Master Resource Book in JEE Main Physics

Question

A solenoid has 2000 turns wound over a length of \(0.30 \mathrm{~m}\). The area of its cross-section is \(1.2 \times 10^{-3} \mathrm{~m}^{2}\). Around its central section, a coil of 300 turns is wound. If an initial current of \(2 \mathrm{~A}\) in the solenoid is reversed in \(0.25 \mathrm{~s}\), then the emf induced in the coil is equal to (a) \(6 \times 10^{-4} \mathrm{~V}\) (b) \(4.8 \times 10^{-2} \mathrm{~V}\) (c) \(6 \times 10^{-2} \mathrm{~V}\) (d) \(48 \mathrm{kV}\)

Step-by-Step Solution

Answer

The induced emf in the coil is \(4.8 \times 10^{-2} \, V\), option (b).

Step 1: Calculate Solenoid's Magnetic Field

The magnetic field inside a solenoid is given by the formula \( B = \mu_0 n I \), where \( n \) is the number of turns per unit length, and \( I \) is the current. Here, \( n = \frac{2000}{0.30} \) and \( I = 2 \mathrm{~A} \). Substituting the values, we get \( B = \mu_0 \times \frac{2000}{0.30} \times 2 \).

Step 2: Compute Magnetic Flux through the Coil

The magnetic flux \( \Phi \) through the coil is given by \( \Phi = B \times A \times N_c \), where \( A \) is the cross-sectional area (\(1.2 \times 10^{-3} \mathrm{~m}^2\)) and \( N_c \) is the number of turns in the coil (300). Substitute the magnetic field from the first step to find \( \Phi \).

Step 3: Determine Change in Magnetic Flux

As the current direction is reversed, the total change in flux \( \Delta \Phi \) is \( 2 \Phi \) since \( I \) goes from +2 A to -2 A. Compute \( \Delta \Phi \) using the flux \( \Phi \) computed in the previous step.

Step 4: Calculate Induced EMF

Using Faraday's law of electromagnetic induction, the induced EMF \( \varepsilon \) is given by \( \varepsilon = -\frac{\Delta \Phi}{\Delta t} \), where \( \Delta t = 0.25 \mathrm{~s} \). Substitute \( \Delta \Phi \) from the previous step to calculate \( \varepsilon \).

Step 5: Select and Verify the Correct Option

Calculate \( \varepsilon \) from the previous step and compare it with the options given. We find \( \varepsilon \approx 4.8 \times 10^{-2} \mathrm{~V} \), which matches option (b).

Key Concepts

Induced EMF

When we talk about Induced Electromotive Force (EMF), we're essentially discussing the voltage generated in a coil or conductor when it experiences a change in its magnetic environment. Imagine it as the electrical pressure that urges currents to flow due to changing magnetic fields. It doesn't work like a battery, but it can still cause current to flow if there's a path to do so.

To get a sense of how induced EMF works, think about a loop of wire in a magnetic field. If the magnetic field around the loop changes, an induced EMF is generated. This happens because the changing magnetic field alters the magnetic flux through the coil, which essentially is the product of the magnetic field strength and the area it encompasses.

This induced EMF is a result of a fundamental principle of physics closely related to the movement and transformation of energy.

Faraday's Law

Faraday's Law is the backbone of electromagnetic induction. It's a principle that quantifies how changing magnetic fields can induce an electric current. According to this law, the magnitude of the induced EMF is proportional to the rate of change of the magnetic flux through the coil. In mathematical terms, it's expressed as:

\[ \varepsilon = -\frac{d\Phi}{dt} \]

Here, \( \varepsilon \) represents the induced EMF, and \( \Phi \) is the magnetic flux. The negative sign tells us about the direction of the induced EMF according to Lenz's Law, which states that the induced EMF opposes the change in flux that produced it.

Faraday's Law helps us understand that to induce a greater EMF, either increase the rate of change of the magnetic flux, or amplify the number of turns the coil has. This explains why tightly wound coils are often used in electrical devices.

Magnetic Flux

Magnetic flux is a concept that helps quantify the quantity of magnetism, taking into account the strength and the extent of a magnetic field. It's what the magnetic field 'flows' through, and its level is a product of the magnetic field strength and the area through which it penetrates, as well as the angle it makes with the surface. Mathematically, it can be expressed as:

\[ \Phi = B \cdot A \cdot \cos \theta \]

In this equation, \( B \) is the magnetic field, \( A \) is the area, and \( \theta \) is the angle between the magnetic field and the perpendicular to the surface. When \( \theta \) is zero (meaning the field and the normal to the surface is perfectly aligned), the formula simplifies to \( \Phi = B \cdot A \).

Understanding magnetic flux is crucial in grasping how EMF is induced in a coil. A change in the magnetic flux (\( \Delta \Phi \)) through a coil directly leads to an induced EMF, as explained by Faraday's Law. It's all about the interaction between fields and the areas they influence, paving the way for energy transformation in magnetic induction.