Two circular coils made of similar wires but of radii 20 and 40cm are connected in parallel. Find the ratio of the magnetic fields at their centers.

Explanation

To find the ratio of the magnetic fields at the centers of two circular coils connected in parallel, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii of the Coils**:
- Let the radius of the first coil \( r_1 = 20 \, \text{cm} = 0.2 \, \text{m} \).
- Let the radius of the second coil \( r_2 = 40 \, \text{cm} = 0.4 \, \text{m} \).

2. **Understand the Resistance of the Coils**:
- Since both coils are made of similar wires, their resistances \( R_1 \) and \( R_2 \) can be expressed in terms of their lengths.
- The length of a circular coil is given by the circumference, \( L = 2\pi r \).
- Therefore, \( R_1 \propto L_1 \) and \( R_2 \propto L_2 \).
- Thus, \( R_1 \propto 2\pi r_1 \) and \( R_2 \propto 2\pi r_2 \).

3. **Calculate the Resistance Ratio**:
- The resistance ratio can be simplified as:
\[ \frac{R_1}{R_2} = \frac{2\pi r_1}{2\pi r_2} = \frac{r_1}{r_2} \]
- Substituting the values:
\[ \frac{R_1}{R_2} = \frac{20 \, \text{cm}}{40 \, \text{cm}} = \frac{1}{2} \]

4. **Determine the Current Ratio**:
- Since the coils are connected in parallel, the current divides inversely with resistance:
\[ \frac{I_1}{I_2} = \frac{R_2}{R_1} = \frac{2}{1} \]

5. **Calculate the Magnetic Field at the Center of Each Coil**:
- The magnetic field at the center of a circular coil is given by:
\[ B = \frac{\mu_0 I}{2\pi r} \]
- Therefore, the magnetic fields for the two coils are:
\[ B_1 = \frac{\mu_0 I_1}{2\pi r_1} \quad \text{and} \quad B_2 = \frac{\mu_0 I_2}{2\pi r_2} \]

6. **Find the Ratio of the Magnetic Fields**:
- The ratio of the magnetic fields can be expressed as:
\[ \frac{B_1}{B_2} = \frac{I_1}{I_2} \cdot \frac{r_2}{r_1} \]
- Substituting the values we have:
\[ \frac{B_1}{B_2} = \left(\frac{2}{1}\right) \cdot \left(\frac{0.4}{0.2}\right) = 2 \cdot 2 = 4 \]

7. **Conclusion**:
- Therefore, the ratio of the magnetic fields at the centers of the two coils is:
\[ \frac{B_1}{B_2} = 4:1 \]

### Final Answer:
The ratio of the magnetic fields at their centers is \( 4:1 \).