Question

A uniform magnetic field $\overrightarrow{B}=0.25\hat{k}\text{T}$ exists in a circular region of radius $R=5\text{m}$. A loop of radius $R=5\text{m}$ lying in x - y plane encloses the magnetic field at $t=0$ and then pulled at uniform velocity $\overrightarrow{v}=4\hat{i}\text{m/s}.$ Find the emf induced (in volts) in the loop at $t=2\text{sec}$.

Answer 1

From above figure.
$D=2R-vt$
$C_{1}M=(vt-R)+\left(\frac{D}{2}\right)$
$C_{1}M=vt-R+\frac{2R-vt}{2}$
$C_{1}M=\frac{vt}{2}$
From $\Delta P{C}_{1}M$
$PQ=2PM=2\sqrt{R^2-(C_{1}M{)}^2}$
$=2\sqrt{R^2-{\left(\frac{vt}{2}\right)}^2}=\sqrt{4R^2-v^2{t}^2}$
Emf induced = emf generated in PQ
Emf $=Bv\left(PQ\right)$
$e=B\times v\times \sqrt{4R^2-v^2{t}^2}=0.25\times 4\times \sqrt{4(5{)}^2-(4{)}^2(2{)}^2}$
$e=\sqrt{100-64}=\sqrt{36}$
$e=6$ volts .