A long straight wire is placed along the axis of a circuit ring of radius R. The mutual inductance of this system is

Options

A. `(mu_(0)R)/2`
B. `(mu_(0)piR)/2`
C. `mu_(0)/2`
D. 0

Answer

To find the mutual inductance of a long straight wire placed along the axis of a circular ring of radius R, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
We have a long straight wire placed along the axis of a circular ring (loop) with radius R. The wire carries a current I.

2. **Magnetic Field Due to the Wire**:
The magnetic field (B) produced by a long straight wire at a distance r from it is given by the formula:
\[ B = \frac{\mu_0 I}{2\pi r} \]
where \(\mu_0\) is the permeability of free space.

3. **Magnetic Flux Through the Ring**:
The magnetic flux (Φ) through the circular ring due to the magnetic field from the wire can be calculated. The area (A) of the ring is given by:
\[ A = \pi R^2 \]
The magnetic field is perpendicular to the area vector of the ring. Therefore, the magnetic flux is:
\[ \Phi = B \cdot A = B \cdot \pi R^2 \]

4. **Substituting the Magnetic Field**:
Substitute the expression for B into the flux equation:
\[ \Phi = \left(\frac{\mu_0 I}{2\pi R}\right) \cdot \pi R^2 = \frac{\mu_0 I R}{2} \]

5. **Calculating Mutual Inductance (M)**:
The mutual inductance (M) is defined as the ratio of the magnetic flux through the ring to the current in the wire:
\[ M = \frac{\Phi}{I} = \frac{\frac{\mu_0 I R}{2}}{I} = \frac{\mu_0 R}{2} \]

6. **Conclusion**:
Therefore, the mutual inductance of the system is:
\[ M = \frac{\mu_0 R}{2} \]

### Final Answer:
The mutual inductance of the system is \( M = \frac{\mu_0 R}{2} \).

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