Question

A uniform, non-time-varying magnetic field of 3 T points 30⁰ away from the perpendicular to the plane of a rectangular loop of wire 0.1 m by 0.2m (Figure 22.28). The loop as a whole is moved in such a way that it maintains its shape and its orientation in the uniform magnetic field. What is the emf around the loop during this move? In 0.1s the loop in Figure 22.28 is stretched to be 0.12m by 0.22 m while keeping the centre of the loop in one place. What is the average emf around the loop during this time?

Short answer

The emf of the loop after 0.1 s is 0.166 V.

Step-by-step solution

A concept:

The emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit.The loop surrounding region A lies in a uniform magnetic field B.

The induced emf is,

$\mathit{\epsilon }\mathbf{=}\mathbf{-}\mathit{B}\mathit{A}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }$

A given data:

The magnetic field, $B=3\text{T}$

The angle, $\theta =30°$

Initial length of the loop. $l_1=0.1\text{m}$

Initial width, $b_1=0.2\text{m}$

Final length,$l_2=0.22\text{m}$

Final width,$b_2=0.12\text{m}$

Time, $t=0.1\text{s}$

The average emf around the loop:

Let the area of the loop is changed from $A_1$to $A_2$. The expression for the emf is,

$emf=-\frac{\left(B\cdot \left(A_1-A_2\right)\right)}{t}$

As the area is the product of the length and the breadth, the above equation becomes.

$\begin{array}{c}emf=\frac{\left(B\cdot \left(l_2{b}_2-l_1{b}_1\right)\right)}{t}\\ =B\frac{\left(l_2{b}_2-l_1{b}_1\right)}{t}\cos \theta \end{array}$

Substitute known values in the above equation.

$\begin{array}{c}emf=\left(3\text{T}\right)\frac{\left(\text{0.22m}\right)\left(\text{0.12m}\right)-\left(\text{0.1m}\right)\left(\text{0.2m}\right)}{0.1\text{s}}\times \cos 30°\\ =3\text{T}\times \frac{\left({\text{0.0264m}}^2\right)-\left({\text{0.02m}}^2\right)}{0.1\text{s}}\times 0.866\\ =3\text{T}\times 0.064{\text{m}}^2\times 0.866\\ =0.166\text{V}\end{array}$

Hence, the emf of the loop after 0.1s is$0.166V$ .