When equal current is passed through two coils equal magnetic field is produced at their centres. 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 when equal currents are passed through them, and the ratio of the number of turns in the coils is given as 8:15.

### Step-by-Step Solution:

1. **Understanding the Magnetic Field in Coils**:
The magnetic field \( B \) at the center of a circular coil carrying current can be expressed as:
\[ B = \frac{N \mu_0 I}{2R} \]
where \( N \) is the number of turns, \( \mu_0 \) is the permeability of free space, \( I \) is the current, and \( R \) is the radius of the coil.

2. **Setting Up the Equations**:
For two coils, we can denote:
- Coil 1: Number of turns \( N_1 \), radius \( R_1 \)
- Coil 2: Number of turns \( N_2 \), radius \( R_2 \)

The magnetic fields at the center of each coil can be expressed as:
\[ B_1 = \frac{N_1 \mu_0 I}{2R_1} \]
\[ B_2 = \frac{N_2 \mu_0 I}{2R_2} \]

3. **Equating the Magnetic Fields**:
Since it is given that the magnetic fields at the centers of both coils are equal:
\[ B_1 = B_2 \]
Therefore,
\[ \frac{N_1 \mu_0 I}{2R_1} = \frac{N_2 \mu_0 I}{2R_2} \]

4. **Cancelling Common Terms**:
We can cancel \( \mu_0 \) and \( I \) from both sides:
\[ \frac{N_1}{R_1} = \frac{N_2}{R_2} \]

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 that the ratio of the number of turns \( N_1 : N_2 = 8 : 15 \). Thus:
\[ \frac{N_1}{N_2} = \frac{8}{15} \]
Therefore:
\[ \frac{R_1}{R_2} = \frac{8}{15} \]

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

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