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 centers coinciding but their planes perpendicular to each other, we can follow these steps:

### Step-by-Step Solution:

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

2. **Magnetic Field at the Center of the Larger Loop**:
- The magnetic field \( B \) at the center of a circular loop carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2b} \]
- Here, \( \mu_0 \) is the permeability of free space.

3. **Magnetic Field Direction**:
- Using the right-hand rule, if the current flows in the larger loop, the magnetic field at the center (point O) will be directed along the axis of the larger loop.

4. **Flux through the Smaller Loop**:
- The magnetic flux \( \Phi \) through the smaller loop is given by:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
- Where \( A \) is the area of the smaller loop and \( \theta \) is the angle between the magnetic field and the area vector of the smaller loop.
- Since the planes of the loops are perpendicular, \( \theta = 90^\circ \), and thus \( \cos(90^\circ) = 0 \).

5. **Calculating the Flux**:
- Therefore, the magnetic flux through the smaller loop becomes:
\[ \Phi = B \cdot A \cdot 0 = 0 \]
- This indicates that there is no magnetic flux linking the smaller loop due to the perpendicular orientation.

6. **Mutual Inductance Calculation**:
- The mutual inductance \( M \) is defined as:
\[ M = \frac{\Phi}{I} \]
- Since the flux \( \Phi \) is zero, we can conclude:
\[ M = \frac{0}{I} = 0 \]

### Final Answer:
The mutual inductance \( M \) for the two loops is \( 0 \).