Question

A uniform magnetic field is restricted within a region of radius r. The magnetic field changes with time at a rate d⃗B/dt . Loop 1 of radius R > r encloses the region r and loop 2 of radius R is outside the region of magnetic field as shown in the figure below. Then the e.m.f. generated is

(1) zero in loop 1 and zero in loop 2

(2) (- d⃗B/dt)πr² in loop1 and (- d⃗B/dt)πr² in loop 2

(3) (- d⃗B/dt)πR² in loop 1 and zero in loop 2

(4) (- d⃗B/dt)πr² in loop 1 and zero in loop 2

Answer 1

Given that,

A uniform magnetic field is restricted within a region of radius r. The magnetic field changes with time at a rate $\dfrac{dB}{dt}$ .

Magnetic field = B

According to figure,

In loop first,

The magnetic field is inside the loop .

In second loop,

The magnetic field is outside the loop.

We need to calculate the magnetic flux due to loop first

Using formula of flux

$\phi_{1}=B\cdots A$

Where, B = magnetic field

A = surface area

Put the value into the formula

$\Phi_{1}=B\pi R^2$

Where, R = radius

We need to calculate the e.m.f

Using formula of e.m.f

$\epsilon_{1}=-\dfrac{d\phi}{dt}$

Put the value of flux

$\epsilon_{1}=-\dfrac{dB}{dt}\pi R^2$

Now, we need to calculate the magnetic flux due to loop second

Using formula of flux

$\phi_{2}=0$

We need to calculate the e.m.f

Using formula of e.m.f

$\epsilon_{2}=-\dfrac{d\phi_{2}}{dt}$

Put the value of flux

$\epsilon_{2}=0$

Hence, The emf generated is $-\dfrac{dB}{dt}\pi R^2$ in loop first and zero in second loop.

(3) is correct option.