Problem 26 - Master Resource Book in JEE Main Physics

Question

A long solenoid with 15 turns per \(\mathrm{cm}\) has a small loop of area \(2.0 \mathrm{~cm}^{2}\) placed inside the solenoid normal to its axis. If the current carried by the solenoid changes steadily from \(2.0 \mathrm{~A}\) to \(4.0 \mathrm{~A}\) in \(0.1 \mathrm{~s}\), what is the induced emf in the loop while the current is changing? [NCERT] (a) \(7.5 \times 10^{6} \mathrm{~V}\) (b) \(8.5 \times 10^{6} \mathrm{~V}\) (c) \(7.5 \times 10^{4} \mathrm{~V}\) (d) \(7.5 \times 10^{5} \mathrm{~V}\)

Step-by-Step Solution

Answer

The magnitude of the induced emf is \(7.5 \times 10^{-5} \ V\), none of the options match this. Ensure calculation accuracy and recheck options.

Step 1: Calculate the change in magnetic field

The magnetic field inside a solenoid is given by the formula: \( B = \mu_0 n I \), where \( \mu_0 \) is the permeability of free space \( (4\pi \times 10^{-7} \ T \cdot m/A) \), \( n \) is the number of turns per unit length, and \( I \) is the current. Given \( n = 1500 \ \text{turns/m} \) (since \( 15 \ \text{turns/cm} = 1500 \ \text{turns/m} \)), we calculate the change in magnetic field as \( \Delta B = \mu_0 n \Delta I \). Since the current changes from 2 A to 4 A, \( \Delta I = 2 \ A \). Hence, \( \Delta B = 4\pi \times 10^{-7} \times 1500 \times 2 \).

Step 2: Calculate the change in magnetic flux

The magnetic flux \( \Phi \) through the loop is given by \( \Phi = B A \), where \( A = 2.0 \ cm^2 = 2.0 \times 10^{-4} \ m^2 \) is the area of the loop. The change in flux \( \Delta \Phi \) due to the change in \( B \) is \( \Delta \Phi = \Delta B \times A \). Using the \( \Delta B \) calculated from Step 1, compute \( \Delta \Phi = (4\pi \times 10^{-7} \times 1500 \times 2) \times 2.0 \times 10^{-4} \).

Step 3: Calculate the induced emf

The induced emf \( \varepsilon \) can be found by using Faraday's law of induction, which states \( \varepsilon = -\frac{\Delta \Phi}{\Delta t} \). Here, \( \Delta t = 0.1 \ s \) is the time over which the current change occurs. Substitute \( \Delta \Phi \) from Step 2 into this formula to find \( \varepsilon = -\frac{4\pi \times 10^{-7} \times 1500 \times 2 \times 2.0 \times 10^{-4}}{0.1} \). Calculate \( \varepsilon \).

Step 4: Calculate the exact numerical result of the induced emf

Compute the numerical value: \( \varepsilon = -\frac{4\pi \times 10^{-7} \times 1500 \times 2 \times 2.0 \times 10^{-4}}{0.1} = -7.5 \times 10^{-5} \ V \). Neglecting the negative sign which indicates the direction of emf, the magnitude of the induced emf is \( 7.5 \times 10^{-5} \ V \).

Step 5: Match the answer with provided options

Check the calculated emf against the given answer choices: (a) \(7.5 \times 10^{6} \ V\), (b) \(8.5 \times 10^{6} \ V\), (c) \(7.5 \times 10^{4} \ V\), (d) \(7.5 \times 10^{5} \ V\). None of these match the correct calculated value. Therefore, it's possible that the provided options are misprints or errors. As per the calculated results, the value should match a magnitude of \(7.5 \times 10^{-5} \ V\).

Key Concepts

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle in physics that explains how electrical currents are generated in conductors when exposed to changing magnetic fields. It's a cornerstone concept in electromagnetism and is essential for understanding how many types of electrical machinery work, such as transformers and electric motors.

**Understanding Faraday's Law**
Faraday's Law states that the induced electromotive force (EMF) in any closed circuit is equal to the negative rate of change of magnetic flux through the circuit. Mathematically, it can be represented as:\[\varepsilon = -\frac{d\Phi}{dt} \],
where \( \varepsilon \) is the induced EMF, \( \Phi \) is the magnetic flux, and \( \frac{d\Phi}{dt} \) is the rate of change of that flux. The negative sign in the equation represents Lenz's law, which states that the EMF will induce a current in a direction that opposes the change in magnetic flux.

In practical terms, if you increase the magnetic field passing through a loop of wire, an EMF will be induced that produces a current in the opposite direction to resist that increase. Consequently, decreasing the magnetic flux will induce a current that attempts to maintain the original magnetic field. This is why Faraday's Law is crucial for such devices as electric generators and induction cooktops.

Magnetic Flux

Magnetic Flux is a measure of the quantity of magnetism, taking into account the strength and the extent of a magnetic field. It is represented as the flow of a magnetic field through a given area, and is mathematically defined as the product of the magnetic field and the area perpendicular to the field that it passes through.

**Defining Magnetic Flux**
The mathematical representation of magnetic flux is given by:\[\Phi = B \cdot A \cdot \cos(\theta) \],
where \( \Phi \) is the magnetic flux, \( B \) is the magnetic field's strength, \( A \) is the area through which the field lines pass, and \( \theta \) is the angle between the magnetic field lines and the perpendicular to the surface.

When the magnetic field is fully perpendicular to the surface, \( \cos(\theta) = 1 \), thereby simplifying the equation to \( \Phi = B \cdot A \). This simplification explains why, in exercises where a loop is oriented normally to the magnetic field (like in our example), the formula for magnetic flux becomes straightforward. Changes in this magnetic flux over time are responsible for inducing EMF, as explained by Faraday's Law.

Solenoid

A Solenoid is a coil of wire that is designed to create a magnetic field when electric current is passed through it. They are commonly used in various electromagnetic devices due to their ability to convert electrical energy into mechanical motion, such as in electromagnets, inductors, or relays.

**Characteristics of a Solenoid**
Solenoids are characterized by several features:

• **Turns Per Unit Length:** The efficacy of a solenoid to create a magnetic field is partly dependent on its number of turns per unit length. More turns per cm or meter generally imply a stronger magnetic field.


• **Length of the Coil:** Long solenoids have a more uniform magnetic field inside them, which is useful when precise and consistent magnetic fields are necessary.


• **Current Flowing Through It:** The strength of the magnetic field generated by a solenoid is directly proportional to the current that flows through it. Increasing the current will strengthen the field and vice versa.

Recognizing the magnetic field in a solenoid can be calculated using \( B = \mu_0 \cdot n \cdot I \), where \( \mu_0 \) is the magnetic permeability of free space, \( n \) is the number of turns per length, and \( I \) is the current, is crucial in understanding how solenoids work and their applications in real-world devices.