A conducting circular loop is placed in a uniform magnetic field, B = 0.025 T with its plane perpendicular to the magnetic field. The radius of the loop is made to shrink at a constant rate of $2\, mm\, s^{-1}$. The induced emf when the radius is 2 cm, is

$2π\, μV$

The correct answer is Option (2) → $2π\, μV$

Given:

$B = 0.025~\text{T}$

Rate of change of radius: $\frac{dr}{dt} = -2~\text{mm/s} = -2 \times 10^{-3}~\text{m/s}$

Radius at the instant: $r = 2~\text{cm} = 0.02~\text{m}$

Magnetic flux through the loop:

$\Phi = B \cdot A = B \cdot \pi r^2$

Induced emf:

$\mathcal{E} = - \frac{d\Phi}{dt} = - \frac{d}{dt} (B \pi r^2) = - B \pi \frac{d(r^2)}{dt}$

$\frac{d(r^2)}{dt} = 2 r \frac{dr}{dt}$

So, $\mathcal{E} = - B \pi (2 r \frac{dr}{dt}) = - 2 \pi B r \frac{dr}{dt}$

Substitute values:

$\mathcal{E} = 2 \pi (0.025)(0.02)(2 \times 10^{-3})$

$\mathcal{E} = 2 \pi \cdot 0.025 \cdot 4 \times 10^{-5}$

$\mathcal{E} = 2 \pi \cdot 10^{-6}$