A big circular coil of 1000 turns and average radius 10 m is rotating about its horizontal diameter at 2 rad s^-1. If the vertical component of earth's magnetic field at that place is 2 × 10^-5  T and electrical resistance of the coil is 12.56 Ω, then the maximum induced current in the coil will be:

Concept:

In this, we have to use the induced emf concept so that we can calculate the maximum current in the coil. 

Induced emf can be explained as It relates to the magnetic flux through the circuit's temporal rate of change directly. Consider a loop that encloses a space. A is located in a steady magnetic field. 

Magnetic flux in coil is: \(\phi = B.Acosθ \)    ----- (1)

Induced emf is = \(\varepsilon = -\frac{d\phi }{dt} = -\frac{d(B.Acosθ )}{dt} \Rightarrow \) B.A (sinθ )(dθ /dt) = B.A sinθ ω 

for maximum induced emf sinθ = 1 then ϵ = BAω 

for N number of turns in coils  ϵ = NBAω   ----- (2)

ω = angular frequency of coil, N = number of coils, A = area of coil , B = magnetic field 

Maximum induced current in the coil is = i_max = \(\frac{\varepsilon_{max}}{R}\)    ----- (3)

where R = electrical resistance of the coil.

Calculation:

Given: 

Number of turns in coils = 1000, magnetic field B = 2× 10^-5 T,   average radius of circular coil = 10 m, 

Angular frequency of coil = 2 rad/s, 

The electrical resistance of the coil is = 12.56 ohms

Using equation (3) we can find the value of maximum induced current in coil =  imax = \(\frac{\varepsilon_{max}}{R}\) 

Putting the value of maximum emf from equation 2 we get: maximum current = NABω /R

i_max = 1000× π × (10)²× 2×2× 10^-5/ 12.56 = 1 A

Hence Option 4) is correct.