Question

Two circular coils A and B are made from the same wire but the radius of coil A is twice that of coil B. If the magnetic fields at their centres are the same, then the ratio of potential differences applied across A to that of B is ______.

Options

• 1 : 4

• 4 : 1

• 2 : 1

• 1 : 2

Answer 1

Two circular coils A and B are made from the same wire but the radius of coil A is twice that of coil B. If the magnetic fields at their centres are the same, then the ratio of potential differences applied across A to that of B is 4 : 1.

Explanation:

According to the question,

The magnetic field at the centre of coil A = Magnetic field at the centre of coil B

`(mu_0l_1)/(2(2r)) = (mu_0l_2)/(2r) ⇒ l_1/l_2 = 2` ........(i)

We know, R = `rho(l/A)`, where ρ is resistivity, l is length and A is an area of cross-section.

⇒ `l_1 = V_1/R_1 = V_1/(rho(l_1/A)) ⇒ V_1/l_1 = rho . l_1/A` .....(ii)

and `l_2 = V_2/R_2 = V_2/(rho . (l_2/A))`

⇒ `V_2/l_2 = rho . l_2/A` ..........(iii)

From Eqs. (ii) and (iii), we get

`V_1/l_1 xx l_2/V_2 = l_1/l_2 = (2pir_1)/(2pir_2)`

⇒ `V_1/V_2 . l_2/l_1 = r_1/r_2 ⇒ V_1/V_2 . l_2/l_1 = (2r)/r`

⇒ `V_1/V_2 = 2l_1/l_2 = 2 xx 2`           [From Eq. (i)]

⇒ `V_1/V_2 = 4/1 = 4 : 1`