AIIMS 2004 Physics Magnetic Field of a Coil MCQ Question

To analyze the magnetic field produced by a circular coil of radius $R$ carrying an electric current $I$ at a point along its axis, we can use the principles of magnetism and the Biot-Savart law.

Explanation of the Correct Answer

When we consider a circular coil with a current flowing through it, the magnetic field $B$ at a point along its axis at a distance $r$ from the center of the coil (where $r \gg R$) can be approximated.

The magnetic field $B$ along the axis of a circular loop of radius $R$ and carrying current $I$ at a distance $z$ from the center of the loop is given by the formula:

$$B(z) = \frac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}$$

In our scenario, where $r$ (the distance along the axis) is much greater than $R$ (i.e., $r \gg R$), we can simplify the expression. Here, we can approximate $R^2 + r^2 \approx r^2$. Hence, the formula simplifies to:

$$B(r) \approx \frac{\mu_0 I R^2}{2r^3}$$

From this equation, we can see that the magnetic field $B$ varies as $\frac{1}{r^3}$ when $r$ is much larger than $R$. Therefore, the correct answer is:

Correct Answer: C) $\frac{1}{r^3}$

(Note: There seems to be a mistake in the question stating the correct answer as D. The correct variation is indeed $\frac{1}{r^3}$.)

Clarification of Other Options

Now, let’s analyze why the other options are incorrect:

• Option A) $\frac{1}{r}$: This would imply that the magnetic field decreases linearly with distance. This is not the case for a coil since magnetic fields from current-carrying loops decrease more rapidly with distance.

• Option B) $\frac{1}{r^2}$: This relationship might be expected for point magnetic dipoles, but for a circular coil at large distances, the magnetic field decreases more rapidly, as derived above.

• Option D) $\frac{1}{r^4}$: This option suggests an even more rapid decrease than what is derived. The field from a coil behaves like that of a magnetic dipole at large distances, which is $\frac{1}{r^3}$, not $\frac{1}{r^4}$.

Conclusion

In conclusion, for a circular coil of radius $R$ carrying current $I$, the magnetic field at a distance $r$ on its axis, when $r \gg R$, varies as $\frac{1}{r^3}$. Therefore, the correct answer is:

C) $\frac{1}{r^3}$.

This illustrates the understanding of how magnetic fields behave in relation to distance and the geometry of current-carrying structures.