Question

The figure shows a rectangular loop of wire immersed in a non-uniform and varying magnetic field $\overrightarrow{B}$ that is perpendicular to and directed into the page. The field's magnitude is given by $B=4t^2{x}^2$, with $B$ in $\text{tesla}$, $t$ in $\text{second}$, and $x$ in $\text{metre}$. The loop has width $W=3.0\text{m}$ and height $H=2.0\text{m}$. What is the magnitude of the induced emf $E$ around the loop at $t=0.10\text{s}$ ?

• A. $12.4\text{V}$
• B. $13.4\text{V}$
• C. $14.4\text{V}$
• D. $15.4\text{V}$

Answer 1

The correct option is C $14.4\text{V}$

Let us assume an element of thickness $dx$ as shown in the figure.



So, its area is $dA=Hdx$.

The magnetic flux through this area,

$d\varphi =\overrightarrow{B}.\overrightarrow{dA}=BdA\cos 0^{\circ }=BdA$

$d\varphi =BHdx=4t^2{x}^{2}Hdx$

The total flux through the entire loop,

$\varphi =\int d\varphi =4t^{2}H{\int }_0^{3.0}{x}^{2}dx$

$=4t^{2}H{\left[\frac{x^3}{3}\right]}_0^{3.0}=72t^2[\because H=2.0\text{m}]$

Now the magnitude of induced emf be,

$E=|-\frac{d\varphi }{dt}|=\frac{d\left(72t^2\right)}{dt}=144t$

So, at $t=0.10\text{s}$,

$E=144\times 0.10=14.4\text{V}$

Hence, $\left(C\right)$ is the correct answer.