JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Electromagnetic Induction

Two concentric circular coils,

C_{1}

and

C_{2}

are placed in the XY plane.

C_{1}

has 500 turns, and a radius of

1

cm.

C_{2}

has 200 turns and radius of

20

cm.

C_{2}

carries a time dependent current

I (t) = (5 t^{2} - 2 t + 3)

A where t is in s. The emf induced in

C_{1}

(in mV), at the instant

t = 1

s is

\frac{4}{x}

. The value of

x

is ______.

Answer: 5

Solution

The inner coil

C_{1}

is small enough to lie in the almost uniform field at the centre of

C_{2}

, namely

B = \frac{μ _{0} N _{2} I}{2 R _{2}}

. The flux it links is

Φ = N_{1} (π R_{1}^{2}) B = \frac{μ _{0} N _{1} N _{2} π R _{1}^{2}}{2 R _{2}} I

, so the induced emf is

ε = \frac{μ _{0} N _{1} N _{2} π R _{1}^{2}}{2 R _{2}} \frac{d I}{d t}

. With

I = 5 t^{2} - 2 t + 3

\frac{d I}{d t} = 10 t - 2

, which is

8

A/s at

t = 1

s. Putting in

N_{1} = 500

N_{2} = 200

R_{1} = 0.01

m and

R_{2} = 0.2

m gives

ε \approx 0.8

mV. Writing this as

\frac{4}{x}

mV fixes

x = 5

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Integer answer