$ ext{The magnetic field at the centre of a circular coil of radius } r ext{ and carrying a current } I ext{ is } B. ext{ What is the magnetic field at a distance } x = \sqrt{3}r ext{ from the centre, on the axis of the coil ?}$

Q: $\text{The magnetic field at the centre of a circular coil of radius } r \text{ and carrying a current } I \text{ is } B. \text{ What is the magnetic field at a distance } x = \sqrt{3}r \text{ from the centre, on the axis of the coil ?}$

$\text{The magnetic field at the centre of a circular coil of radius } r \text{ and carrying a current } I \text{ is } B. \text{ What is the magnetic field at a distance } x = \sqrt{3}r \text{ from the centre, on the axis of the coil ?}$

Question

$	ext{The magnetic field at the centre of a circular coil of radius } r 	ext{ and carrying a current } I 	ext{ is } B. 	ext{ What is the magnetic field at a distance } x = \sqrt{3}r 	ext{ from the centre, on the axis of the coil ?}$

Accepted Answer

To determine the magnetic field at a distance $ x = \sqrt{3}r $ from the center, on the axis of a circular coil, we need to use the formula for the magnetic field along the axis of a circular loop. The magnetic field $ B_x $ at a distance $ x $ from the center of a circular coil of radius $ r $ carrying a current $ I $ is given by:

$$
B_x = \frac{\mu_0 I r^2}{2 (r^2 + x^2)^{3/2}}
$$

where $ \mu_0 $ is the permeability of free space.

1. **Magnetic Field at the Center:**
   At the center of the coil ($ x = 0 $), the magnetic field $ B $ is given by:

$$
   B = \frac{\mu_0 I}{2r}
   $$

2. **Magnetic Field at $ x = \sqrt{3}r $:**
   Substituting $ x = \sqrt{3}r $ into the formula for $ B_x $:

$$
   B_x = \frac{\mu_0 I r^2}{2 (r^2 + (\sqrt{3}r)^2)^{3/2}}
   $$

Simplifying the expression inside the parentheses:

$$
   r^2 + (\sqrt{3}r)^2 = r^2 + 3r^2 = 4r^2
   $$

Therefore:

$$
   B_x = \frac{\mu_0 I r^2}{2 (4r^2)^{3/2}} = \frac{\mu_0 I r^2}{2 (8r^3)} = \frac{\mu_0 I}{16r}
   $$

Comparing this with the magnetic field at the center $ B = \frac{\mu_0 I}{2r} $, we find:

$$
   B_x = \frac{B}{8}
   $$

Thus, the magnetic field at a distance $ x = \sqrt{3}r $ from the center, on the axis of the coil, is $ \frac{B}{8} $.

3. **Explanation of Options:**

- **Option 1 (B):** This option suggests that the magnetic field at $ x = \sqrt{3}r $ is the same as at the center, which is incorrect because the field decreases with distance along the axis.
   
   - **Option 2 (B/2):** This option implies that the field is half of that at the center, which is not supported by the calculations.
   
   - **Option 3 (8B):** This option suggests an increase in the magnetic field, which contradicts the nature of the field decreasing with distance from the center.
   
   - **Option 4 (B/8):** This is the correct option as derived from the calculations above, showing that the magnetic field at $ x = \sqrt{3}r $ is indeed $ \frac{B}{8} $.

Therefore, the correct answer is Option 4, $ \frac{B}{8} $.