Two circular coils 1 and 2 are made from the same wire but the radius of the 1st coil is twice that of the 2nd coil. What potential difference in volts should be applied across them so that the magnetic field at the centres is the same.

• (A) 4 times of first coil
• (B) 6 times of first coil
• (C) 3 times of first coil
• (D) 2 times of first coil

To determine the potential difference required for the magnetic fields at the centers of two circular coils to be the same, we need to consider the relationship between the radius of the coils and the magnetic field produced. The magnetic field \( B \) at the center of a circular coil is given by the formula:

Magnetic Field Formula

The magnetic field \( B \) at the center of a coil is proportional to the current \( I \) and the number of turns \( N \), and inversely proportional to the radius \( r \) of the coil:

B = \frac{{\mu_0 N I}}{{2r}}

Coil Specifications

• Let the radius of coil 2 be \( r \).
• Then, the radius of coil 1 is \( 2r \).

Magnetic Field Comparison

For coil 1:

B_1 = \frac{{\mu_0 N_1 I_1}}{{4r}}

For coil 2:

B_2 = \frac{{\mu_0 N_2 I_2}}{{2r}}

Setting the Magnetic Fields Equal

To have the same magnetic field at the centers:

B_1 = B_2

Substituting the expressions:

\frac{{\mu_0 N_1 I_1}}{{4r}} = \frac{{\mu_0 N_2 I_2}}{{2r}}

Solving for Current

This simplifies to:

N_1 I_1 = 2 N_2 I_2

Potential Difference Relation

The potential difference \( V \) across a coil is related to the current and resistance. If both coils are made from the same wire, their resistances will be proportional to their radii. Thus, we can express the potential differences as:

V_1 = I_1 R_1 and V_2 = I_2 R_2

Since \( R_1 \) is twice \( R_2 \) (due to the radius relationship), we find:

V_1 = I_1 (2R_2) and V_2 = I_2 R_2

Final Calculation

To maintain equal magnetic fields, we find that:

V_2 = 2 V_1

Thus, the potential difference across coil 2 should be:

V_2 = 2 \times V_1

Answer

The correct option is (D) 2 times of the first coil.