biot-savart law

Question

A circular coil of radius R carries a current I. The ratio of magnetic field at the centre to the field at a distance R from the centre along the axis is:

RankGuru · Accepted Answer

1. **Field at Center ($B_c$)**: For a circular coil, the magnetic field at the center is $B_c = \frac{\mu_0 I}{2R}$.
2. **Field on Axis ($B_a$)**: The field at a distance $x$ along the axis is $B_a = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$.
3. **Substitute $x = R$**: 
   $$B_a = \frac{\mu_0 I R^2}{2(R^2 + R^2)^{3/2}} = \frac{\mu_0 I R^2}{2(2R^2)^{3/2}} = \frac{\mu_0 I R^2}{2 \cdot 2\sqrt{2} R^3} = \frac{\mu_0 I}{4\sqrt{2}R}$$
4. **Ratio Calculation**: $\frac{B_c}{B_a} = \frac{\mu_0 I / 2R}{\mu_0 I / 4\sqrt{2}R} = 2\sqrt{2}$.

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biot-savart law

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