What is the magnetic field at a distance R from a coil of radius r carrying current l ?

Options

A. `(mu_(0)lR^(2))/(2(R^(2)+r^(2))^((3)/(2)))`
B. `(mu_(0)lr^(2))/(2(R^(2)+r^(2))^((3)/(2)))`
C. `(mu_(0)l)/(2r)`
D. `(mu_(0)l)/(2R)`

Explanation

To find the magnetic field at a distance \( R \) from a coil of radius \( r \) carrying a current \( I \), we can use the formula for the magnetic field due to a circular loop of wire.

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a circular coil of radius \( r \) carrying a current \( I \).
- We want to find the magnetic field \( B \) at a distance \( R \) from the center of the coil along its axis.

2. **Use the Magnetic Field Formula**:
- The magnetic field \( B \) at a distance \( R \) from the center of a circular coil of radius \( r \) carrying a current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I r^2}{2(R^2 + r^2)^{3/2}} \]
- Here, \( \mu_0 \) is the permeability of free space (approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \)).

3. **Substitute Values**:
- If you have specific values for \( I \), \( r \), and \( R \), you can substitute them into the formula to calculate \( B \).
- For example, if \( I = 5 \, \text{A} \), \( r = 0.1 \, \text{m} \), and \( R = 0.2 \, \text{m} \):
\[ B = \frac{(4\pi \times 10^{-7}) \times 5 \times (0.1)^2}{2((0.2)^2 + (0.1)^2)^{3/2}} \]

4. **Calculate the Denominator**:
- First, calculate \( R^2 + r^2 \):
\[ R^2 + r^2 = (0.2)^2 + (0.1)^2 = 0.04 + 0.01 = 0.05 \]
- Then calculate \( (0.05)^{3/2} \):
\[ (0.05)^{3/2} = \sqrt{0.05^3} = \sqrt{0.000125} \approx 0.01118 \]

5. **Final Calculation**:
- Substitute back into the formula to find \( B \):
\[ B = \frac{(4\pi \times 10^{-7}) \times 5 \times 0.01}{2 \times 0.01118} \approx \frac{(4\pi \times 10^{-7}) \times 0.05}{0.02236} \]
- Calculate the numerical value to find \( B \).

### Final Answer:
The magnetic field \( B \) at a distance \( R \) from the coil is given by:
\[ B = \frac{\mu_0 I r^2}{2(R^2 + r^2)^{3/2}} \]