Question

A coil having area $2{\mathrm{m}}^2$ is placed in a magnetic field which changes from $1\mathrm{wb}/{\mathrm{m}}^2$ to $4\mathrm{wb}/{\mathrm{m}}^2$ in an interval of 2 second. The emf induced in the coil of single turn is.... (a) $4\mathrm{v}$ (b) $3\mathrm{v}$ (c) $1.5\mathrm{v}$ (d) $2\mathrm{v}$

Short answer

The short answer for the given problem is (b) $3 \mathrm{v}$.

Step-by-step solution

Find the initial and final magnetic flux

To find the initial and final magnetic flux, we need to multiply the magnetic field strength with the area of the coil. The magnetic field strength changes from $1\mathrm{wb}/{\mathrm{m}}^2$ to $4\mathrm{wb}/{\mathrm{m}}^2$. Let ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ be the initial and final magnetic flux, respectively. For the initial magnetic flux: ${\mathrm{\Phi }}_1=B_1\cdot A$ ${\mathrm{\Phi }}_1=(1\mathrm{wb}/{\mathrm{m}}^2)(2{\mathrm{m}}^2)=2\mathrm{wb}$ For the final magnetic flux: ${\mathrm{\Phi }}_2=B_2\cdot A$ ${\mathrm{\Phi }}_2=(4\mathrm{wb}/{\mathrm{m}}^2)(2{\mathrm{m}}^2)=8\mathrm{wb}$

Calculate the change in magnetic flux

Next, we need to find the change in magnetic flux, which is the difference between the final magnetic flux and the initial magnetic flux. Change in magnetic flux: $\mathrm{\Delta }\mathrm{\Phi }={\mathrm{\Phi }}_2-{\mathrm{\Phi }}_1$ $\mathrm{\Delta }\mathrm{\Phi }=8\mathrm{wb}-2\mathrm{wb}=6\mathrm{wb}$

Calculate the induced emf using Faraday's law

Now, we can calculate the induced emf using Faraday's law of electromagnetic induction. The formula is: $|emf|=\frac{|\mathrm{\Delta }\mathrm{\Phi }|}{\mathrm{\Delta }t}$ where $|emf|$ is the induced emf, $|\mathrm{\Delta }\mathrm{\Phi }|$ is the change in magnetic flux, and $\mathrm{\Delta }t$ is the time interval. We have $|\mathrm{\Delta }\mathrm{\Phi }|=6\mathrm{wb}$ and $\mathrm{\Delta }t=2\mathrm{s}$. Plugging these values into the formula: $|emf|=\frac{6\mathrm{wb}}{2\mathrm{s}}$ $|emf|=3\mathrm{v}$

Choose the correct option

The induced emf in the single-turn coil is calculated as 3 volts. Therefore, the correct answer is: (b) 3 volts

Key concepts

Magnetic Flux

Magnetic flux is a measure of the magnetic field passing through a given surface. It provides an understanding of how much magnetic field "flows" through an area. Imagine it as lines of magnetic force. The more lines that pass through a surface, the greater the magnetic flux. Mathematically, magnetic flux $\mathrm{\Phi }$ is calculated using the formula $\mathrm{\Phi }=B\cdot A\cdot \cos (\theta )$, where $B$ is the magnetic field strength, $A$ is the area it passes through, and $\theta$ is the angle between the magnetic field and the normal to the surface.

In simpler cases, such as the problem at hand where the magnetic field is perpendicular to the coil, the formula simplifies to $\mathrm{\Phi }=B\cdot A$. Here, the coil area is given as 2 ${\mathrm{m}}^2$, with the magnetic field changing from 1 ${\mathrm{wb}/\mathrm{m}}^2$ to 4 ${\mathrm{wb}/\mathrm{m}}^2$. Therefore, the initial magnetic flux is $2\mathrm{wb}$ and the final magnetic flux is $8\mathrm{wb}$. This change in magnetic flux is an important factor in understanding electromagnetic induction.

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle that describes how voltage (or electromotive force, emf) is induced in a circuit due to a change in magnetic flux. Michael Faraday discovered this relationship, showing that a change in magnetic environment of a coil of wire induces a voltage in the coil.

This phenomenon is captured by Faraday's Law, which states that the induced emf in any closed circuit is equal to the rate of change of magnetic flux through the circuit. Mathematically, it is expressed as $|emf|=\left|\frac{\mathrm{\Delta }\mathrm{\Phi }}{\mathrm{\Delta }t}\right|$, where $\mathrm{\Delta }\mathrm{\Phi }$ is the change in magnetic flux and $\mathrm{\Delta }t$ is the change in time.

In the context of the exercise, we calculated the change in magnetic flux as $6\mathrm{wb}$ over a time interval of 2 seconds. Using Faraday's Law, this results in an induced emf of $3\mathrm{v}$.

Induced EMF

The concept of induced EMF is central to electromagnetic induction, a crucial process in electromagnetism. An electromotive force (emf) is generated in a conductor when it experiences a change in magnetic field.

This induced emf arises due to the movement of electrons in the conductor, driven by the changing magnetic flux. This change can occur by varying the magnetic field strength, the area of the loop, or the angle between the magnetic field and the loop. It can even happen by moving the loop relative to the magnetic field.

Induced emf is the operating principle behind many electrical generators and transformers. It allows for the conversion of mechanical energy into electrical energy, paving the way for the production of electricity.

In the problem, the fluctuating magnetic field through a single-turn coil induces an emf of 3 volts, as calculated using Faraday's Law. This concept is fundamental in understanding not just simple circuits but also complex electric machines.