Question

A circular coil of radius \(R\) carries an electric current \(I\). The magnetic field due to the coil at a point on the axis of the coil located at a distance \(r\) from the centre of the coil, such that \(r \gg R\), the magnetic field at that point is proportional to

• A. \( \dfrac{1}{r^3} \)
• B. \( \dfrac{1}{r} \)
• C. \( \dfrac{1}{r^4} \)
• D. \( \dfrac{1}{r^2} \)

Hint:

At large distances, a current loop behaves like a magnetic dipole, and its field varies as \(1/r^3\).

Answer 1

The Correct Option is A

Step 1: Magnetic field on the axis of a circular coil.
The magnetic field at a point on the axis of a circular current loop is given by \[ B = \frac{\mu_0 I R^2}{2(r^2 + R^2)^{3/2}} \]

Step 2: Apply the condition \( r \gg R \).
When \( r \gg R \), we can approximate \[ (r^2 + R^2)^{3/2} \approx r^3 \]

Step 3: Simplify the expression.
\[ B \propto \frac{1}{r^3} \]