A circular coil 'A' has a radius R and the current flowing through it is I.Another circular coil 'B' has a radius 2R and if `2I` is the current flowing through it then the magnetic fields at the centre of the circular coil are in the ratio of

Options

A. 0.16736111111111
B. 0.084027777777778
C. 0.12569444444444
D. 0.042361111111111

Answer

To find the ratio of the magnetic fields at the center of two circular coils, we can use the formula for the magnetic field at the center of a circular coil, which is given by:

\[ B = \frac{\mu_0 I}{2R} \]

where:
- \( B \) is the magnetic field,
- \( \mu_0 \) is the permeability of free space (a constant),
- \( I \) is the current flowing through the coil,
- \( R \) is the radius of the coil.

### Step 1: Calculate the magnetic field for coil A
For coil A, we have:
- Radius \( R_A = R \)
- Current \( I_A = I \)

Using the formula:
\[ B_A = \frac{\mu_0 I_A}{2R_A} = \frac{\mu_0 I}{2R} \]

### Step 2: Calculate the magnetic field for coil B
For coil B, we have:
- Radius \( R_B = 2R \)
- Current \( I_B = 2I \)

Using the formula:
\[ B_B = \frac{\mu_0 I_B}{2R_B} = \frac{\mu_0 (2I)}{2(2R)} = \frac{\mu_0 I}{2R} \]

### Step 3: Find the ratio of the magnetic fields
Now, we can find the ratio of the magnetic fields \( \frac{B_A}{B_B} \):
\[ \frac{B_A}{B_B} = \frac{\frac{\mu_0 I}{2R}}{\frac{\mu_0 I}{2R}} = 1 \]

Thus, the ratio of the magnetic fields at the center of the two coils is:
\[ B_A : B_B = 1 : 1 \]

### Conclusion
The magnetic fields at the center of the two coils are equal, hence the ratio is \( 1 : 1 \).