Radii of two conducting circular loops are b and a respectively, where `bgtgta` Centres of both loops coincide but planes of both loops are perpendicular each other. The value of mutual inductance for these loops

Options

A. `(mu_0pib^2)/(2a)`
B. Zero
C. `(mu_0piab)/(2(a+b))`
D. `(mu_0pia^2)/(2b)`

Answer

To find the mutual inductance between two conducting circular loops with radii \( b \) and \( a \) (where \( b \gg a \)), and with their planes perpendicular to each other, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have two circular loops: one with radius \( b \) and the other with radius \( a \).
- The centers of both loops coincide, but their planes are perpendicular to each other.

2. **Magnetic Field Due to the Larger Loop**:
- The magnetic field \( B \) at the center of the larger loop (radius \( b \)) due to a current \( I \) flowing through it is given by the formula:
\[ B = \frac{\mu_0 I}{2b} \]
where \( \mu_0 \) is the permeability of free space.

3. **Calculating the Magnetic Flux through the Smaller Loop**:
- The area vector of the smaller loop (radius \( a \)) is perpendicular to its plane. Since the planes of the loops are perpendicular, the angle \( \theta \) between the magnetic field \( B \) and the area vector \( A \) of the smaller loop is \( 90^\circ \).
- The magnetic flux \( \Phi \) through the smaller loop is given by:
\[ \Phi = B \cdot A = B \cdot A \cos(\theta) \]
- Since \( \theta = 90^\circ \), we have:
\[ \Phi = B \cdot A \cos(90^\circ) = 0 \]

4. **Mutual Inductance Calculation**:
- The mutual inductance \( M \) is defined as the ratio of the magnetic flux \( \Phi \) through one loop due to the current in the other loop:
\[ M = \frac{\Phi}{I} \]
- Since we found that \( \Phi = 0 \), we can conclude:
\[ M = \frac{0}{I} = 0 \]

### Conclusion:
The mutual inductance \( M \) between the two loops is zero because the magnetic field lines from the larger loop do not pass through the smaller loop due to the perpendicular orientation of their planes.