An electric current is flowing through a circular coil of radius R. The ratio of the magnetic field at the centre of the coil and that at a distance $\sqrt{3}$R from the centre of the coil on its axis is :

Question

Accepted Answer

To determine the ratio of the magnetic field at the center of a circular coil to the magnetic field at a distance $\sqrt{3}R$ from the center on its axis, we need to use the formula for the magnetic field due to a circular coil.

### Explanation:

1. **Magnetic Field at the Center of the Coil:**
   The magnetic field at the center of a circular coil carrying current $I$ is given by:
   $$
   B_{\text{center}} = \frac{\mu_0 I}{2R}
   $$
   where $\mu_0$ is the permeability of free space, $I$ is the current, and $R$ is the radius of the coil.

2. **Magnetic Field on the Axis of the Coil:**
   The magnetic field at a distance $x$ from the center on the axis of the coil is given by:
   $$
   B_{\text{axis}} = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}
   $$
   For $x = \sqrt{3}R$, this becomes:
   $$
   B_{\text{axis}} = \frac{\mu_0 I R^2}{2(R^2 + (\sqrt{3}R)^2)^{3/2}} = \frac{\mu_0 I R^2}{2(4R^2)^{3/2}} = \frac{\mu_0 I R^2}{2(8R^3)} = \frac{\mu_0 I}{16R}
   $$

3. **Ratio of Magnetic Fields:**
   The ratio of the magnetic field at the center to that at a distance $\sqrt{3}R$ on the axis is:
   $$
   \text{Ratio} = \frac{B_{\text{center}}}{B_{\text{axis}}} = \frac{\frac{\mu_0 I}{2R}}{\frac{\mu_0 I}{16R}} = \frac{16}{2} = 8:1
   $$

### Justification for Option 3:

- **Option 3 (8:1)** is correct because the calculated ratio of the magnetic field at the center to that at a distance $\sqrt{3}R$ on the axis is indeed 8:1, as derived above.

### Explanation for Incorrect Options:

- **Option 1 (2:1):** This option is incorrect because the calculated ratio is 8:1, not 2:1. The magnetic field decreases more significantly along the axis than this option suggests.

- **Option 2 (5:3):** This option is incorrect because the ratio does not simplify to 5:3. The actual ratio is 8:1, which is significantly different.

- **Option 4 (3:2):** This option is incorrect because the ratio of 3:2 does not match the calculated ratio of 8:1. The magnetic field at the center is much stronger compared to the field at $\sqrt{3}R$ on the axis than this option implies.

In conclusion, the correct answer is Option 3 (8:1), as it accurately represents the ratio of the magnetic fields based on the given conditions and formulas.