Problem 22 - Master Resource Book in JEE Main Physics

Question

A coil has an area of \(0.05 \mathrm{~cm}^{2}\) and it has 800 turns. is placed perpendicularly in a magnetic field strength \(4 \times 10^{-5} \mathrm{~Wb} / \mathrm{m}^{2}\), it is rotated through \(90^{\circ} \mathrm{i}\) \(0.1 \mathrm{~s}\). The average emf induced in the coil is (a) \(0.016 \mathrm{~V}\) (b) \(0.032 \mathrm{~V}\) (c) \(0.064 \mathrm{~V}\) (d) \(0.029 \mathrm{~V}\)

Step-by-Step Solution

Answer

(a) 0.016 V

Step 1: Understanding the problem

We are tasked with finding the average electromotive force (emf) induced when a coil with a given area and number of turns rotates in a magnetic field strength for a specified period. The initial position is perpendicular, meaning that the angle is 0 degrees, and it is finally rotated to 90 degrees.

Step 2: Use Faraday's Law of Electromagnetic Induction

Faraday's Law states that the induced emf \(\epsilon\) is given by the negative rate of change of magnetic flux through the coil. Mathematically, it is expressed as \(\epsilon = -N \frac{\Delta \Phi}{\Delta t}\), where \(N\) is the number of turns, \(\Delta \Phi\) is the change in magnetic flux, and \(\Delta t\) is the time duration.

Step 3: Calculate the change in magnetic flux

The magnetic flux \(\Phi\) through a single loop is given by \(\Phi = B \cdot A \cdot \cos(\theta)\), where \(B\) is the magnetic field, \(A\) is the area, and \(\theta\) is the angle between the field and the normal to the loop. Initially, \(\theta_1 = 0^{\circ}\) and finally, \(\theta_2 = 90^{\circ}\). Thus: \[\Delta \Phi = B \cdot A \cdot (\cos(\theta_1) - \cos(\theta_2)) = 4 \times 10^{-5} \times 0.05 \times (\cos(0) - \cos(90))\]Substituting the values into the equation, we get:\[\Delta \Phi = 4 \times 10^{-5} \times 0.05 \times (1 - 0) = 2 \times 10^{-6} \; \text{Wb}\].

Step 4: Calculate the average induced emf

Substitute \(\Delta \Phi\), number of turns \(N\), and the time interval \(\Delta t\) into Faraday's Law to find \(\epsilon\):\[\epsilon = -800 \times \frac{2 \times 10^{-6}}{0.1} = - 0.016 \; \text{V}\]We are usually interested in the magnitude, so the average emf is \(0.016 \; \text{V}\).

Step 5: Select the correct answer

Compare the calculated emf with the options given in the problem. The option that matches our calculation is \(0.016 \; \text{V}\).

Key Concepts

Magnetic Flux

Magnetic flux is a measure of the number of magnetic field lines passing through an area. Imagine it as invisible lines of force flowing across a surface. To calculate magnetic flux, use the formula: \[ \Phi = B \cdot A \cdot \cos(\theta) \] where:

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

Magnetic flux is measured in webers (Wb). Instead of seeing it as a number, think of it as the field's ability to "cut" through an area. The cos(\(\theta\)) part helps us understand the angle's impact: if the field is perpendicular to the area (\(0^\circ\)), all lines pass through, but at \(90^\circ\), no field lines pass through. Let's make it concrete. If you have a tiny rectangular window and a blowing breeze, the breeze represents the magnetic field. If the window stands straight against the wind, it "captures" more breeze, similar to a magnetic field pushing through a perpendicular coil.
So, changes in this flux over time are what bring Faraday's Law into play.

Induced EMF

When there's a change in magnetic flux in a coil, an electromotive force (emf) is induced. This concept is known as Faraday's Law of Electromagnetic Induction. According to this law, \[ \epsilon = -N \frac{\Delta \Phi}{\Delta t} \]where:

• \(\epsilon\) is the induced emf
• \(N\) is the number of turns in the coil
• \(\Delta \Phi\) is the change in magnetic flux
• \(\Delta t\) is the time over which the change occurs

The negative sign indicates that the induced emf acts in a direction to oppose the change in flux—a phenomenon known as Lenz's Law. Think of the emf as a response to a change. If the magnetic environment of a coil changes, the coil "fights back" by inducing an emf to oppose the disturbance. Imagine spinning a bicycle wheel in a static field magnet. As you spin the wheel, the area of loops facing the field changes, causing the magnetic flux through the wheel to change. This change induces an emf, like trying to stop the wheel from spinning.
That's the marvel of induction at work! It's how generators produce electricity—a vital concept for modern technology.

Coil and Magnetic Field Interaction

The interaction between a coil and a magnetic field is foundational to understanding how motors and generators operate. A coil placed in a magnetic field will experience changes in magnetic flux when moved or rotated. In our exercise, the coil rotates from an initial angle of \(0\) degrees to \(90\) degrees relative to the magnetic field lines, altering the magnetic flux. Let's break this down:

• Initially, the coil is at \(0^\circ\), meaning the magnetic field passes perpendicularly through the coil. This maximizes magnetic flux.
• As the coil rotates to \(90^\circ\), the angle reduces the amount of magnetic field passing through it, decreasing the magnetic flux to zero.
• It's this change—from maximum to zero flux—that induces an emf based on Faraday's Law.

Consider how cranes lift and drop loads. As the magnetic grip (like our coil) moves or rotates through fields of influence (the magnetic field), it undergoes varying magnetic interactions. These interactions are precisely what make the mechanisms within generators and other electromagnetic devices work, creating a bridge from physics to real-world applications.
So, whenever a coil moves or a field strength fluctuates, the interactions ensure that energy is efficiently transferred and transformed.