Two circulat coils of radii ratio 1: 2 and turn ratio 4: 1, respectively, are connected in series, The ratio of value of magnetic field at their centre is

Answer

To find the ratio of the magnetic fields at the center of two circular coils with given radii and turns, we can follow these steps:

### Step 1: Understand the formula for the magnetic field at the center of a circular coil
The magnetic field \( B \) at the center of a circular coil is given by the formula:
\[ B = \frac{\mu_0 N I}{2 R} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( N \) is the number of turns,
- \( I \) is the current flowing through the coil,
- \( R \) is the radius of the coil.

### Step 2: Define the parameters for the two coils
Let:
- For Coil 1:
- Radius \( R_1 \)
- Number of turns \( N_1 = 4x \) (since the turn ratio is 4:1)

- For Coil 2:
- Radius \( R_2 = 2R_1 \) (since the radius ratio is 1:2)
- Number of turns \( N_2 = x \)

### Step 3: Write the expressions for the magnetic fields
Using the formula for the magnetic field, we can write:
- For Coil 1:
\[ B_1 = \frac{\mu_0 N_1 I}{2 R_1} = \frac{\mu_0 (4x) I}{2 R_1} = \frac{2 \mu_0 x I}{R_1} \]

- For Coil 2:
\[ B_2 = \frac{\mu_0 N_2 I}{2 R_2} = \frac{\mu_0 x I}{2 (2R_1)} = \frac{\mu_0 x I}{4 R_1} \]

### Step 4: Calculate the ratio of the magnetic fields
Now, we can find the ratio \( \frac{B_1}{B_2} \):
\[ \frac{B_1}{B_2} = \frac{\frac{2 \mu_0 x I}{R_1}}{\frac{\mu_0 x I}{4 R_1}} = \frac{2 \mu_0 x I}{R_1} \cdot \frac{4 R_1}{\mu_0 x I} \]
This simplifies to:
\[ \frac{B_1}{B_2} = 2 \cdot 4 = 8 \]

### Step 5: Conclusion
Thus, the ratio of the magnetic fields at the center of the two coils is:
\[ \frac{B_1}{B_2} = 8 \]