The approximate formula expressing the formula of mutual inductance of two coaxial loops of the same redius `a` when their centers are separated by a distance `l` with `l gt gt a` is

Options

A. `(1)/(2) (mu_(0)pia^(4))/(l^(3))`
B. `(1)/(2) (mu_(0)a^(4))/(l^(2))`
C. `(mu_(0))/(4pi) (pia^(4))/(l^(2))`
D. `(mu_(0))/(pi) (a^(4))/(l^(3))`

Explanation

To find the approximate formula for the mutual inductance \( M \) of two coaxial loops of the same radius \( a \) when their centers are separated by a distance \( l \) (with \( l \gg a \)), we can follow these steps:

### Step 1: Understanding the Magnetic Field Due to One Loop
The magnetic field \( B \) at a distance \( l \) from the center of a single loop carrying a current \( I \) is given by the formula:

\[ B = \frac{\mu_0 I}{4\pi} \cdot \frac{2\pi a^2}{(l^2 + a^2)^{3/2}} \]

Where:
- \( \mu_0 \) is the permeability of free space,
- \( a \) is the radius of the loop,
- \( l \) is the distance from the center of the loop.

### Step 2: Calculate the Magnetic Flux Through the Second Loop
The magnetic flux \( \Phi \) through the second loop due to the magnetic field from the first loop is given by:

\[ \Phi = B \cdot A \]

Where \( A \) is the area of the second loop, which is \( \pi a^2 \). Substituting the expression for \( B \):

\[ \Phi = \left( \frac{\mu_0 I}{4\pi} \cdot \frac{2\pi a^2}{(l^2 + a^2)^{3/2}} \right) \cdot \pi a^2 \]

### Step 3: Simplifying the Expression for Flux
Now we can simplify the expression for magnetic flux:

\[ \Phi = \frac{\mu_0 I}{4\pi} \cdot \frac{2\pi a^2 \cdot \pi a^2}{(l^2 + a^2)^{3/2}} \]

This simplifies to:

\[ \Phi = \frac{\mu_0 I \pi a^4}{2 (l^2 + a^2)^{3/2}} \]

### Step 4: Mutual Inductance Definition
The mutual inductance \( M \) is defined as the ratio of the magnetic flux through one loop to the current in the other loop:

\[ M = \frac{\Phi}{I} \]

Substituting the expression for \( \Phi \):

\[ M = \frac{\frac{\mu_0 I \pi a^4}{2 (l^2 + a^2)^{3/2}}}{I} \]

### Step 5: Final Expression for Mutual Inductance
The \( I \) cancels out, leading to:

\[ M = \frac{\mu_0 \pi a^4}{2 (l^2 + a^2)^{3/2}} \]

### Step 6: Approximation for Large \( l \)
Since \( l \gg a \), we can approximate \( (l^2 + a^2)^{3/2} \approx l^3 \). Therefore, we can simplify \( M \):

\[ M \approx \frac{\mu_0 \pi a^4}{2 l^3} \]

### Final Result
Thus, the approximate formula for the mutual inductance \( M \) of two coaxial loops of the same radius \( a \) when their centers are separated by a distance \( l \) (with \( l \gg a \)) is:

\[ M \approx \frac{\mu_0 \pi a^4}{2 l^3} \]

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