When equal current is passed through two coils equal magnetic field is produced at their centers. If the ratio of number of turns in the coils is 8:15 then the ratio of the their radii will be

Options

A. `1:1`
B. `15:8`
C. `8:15`
D. `1:2`

Answer

To solve the problem, we need to find the ratio of the radii of two coils given that they produce equal magnetic fields at their centers when equal currents are passed through them. We are also given the ratio of the number of turns in the coils.

### Step-by-Step Solution:

1. **Understanding the Magnetic Field Formula**:
The magnetic field \( B \) at the center of a coil is given by the formula:
\[ B = \frac{N \mu_0 I}{2R} \]
where:
- \( N \) = number of turns in the coil
- \( \mu_0 \) = permeability of free space (a constant)
- \( I \) = current through the coil
- \( R \) = radius of the coil

2. **Setting Up the Equations**:
For the first coil, the magnetic field \( B_1 \) can be expressed as:
\[ B_1 = \frac{N_1 \mu_0 I}{2R_1} \]
For the second coil, the magnetic field \( B_2 \) is:
\[ B_2 = \frac{N_2 \mu_0 I}{2R_2} \]

3. **Equating the Magnetic Fields**:
Since the problem states that equal currents are passed through both coils and they produce equal magnetic fields, we can set \( B_1 = B_2 \):
\[ \frac{N_1 \mu_0 I}{2R_1} = \frac{N_2 \mu_0 I}{2R_2} \]

4. **Canceling Common Terms**:
We can cancel \( \mu_0 \) and \( I \) from both sides, as they are common:
\[ \frac{N_1}{2R_1} = \frac{N_2}{2R_2} \]

5. **Rearranging the Equation**:
Rearranging gives us:
\[ \frac{N_1}{N_2} = \frac{R_1}{R_2} \]

6. **Substituting the Given Ratio**:
We are given the ratio of the number of turns \( \frac{N_1}{N_2} = \frac{8}{15} \). Therefore, we can substitute this into our equation:
\[ \frac{8}{15} = \frac{R_1}{R_2} \]

7. **Conclusion**:
Thus, the ratio of the radii \( R_1 : R_2 \) is:
\[ R_1 : R_2 = 8 : 15 \]

### Final Answer:
The ratio of the radii of the two coils is \( 8 : 15 \).