Question

A circular current-carrying coil has radius \( R \). At what distance from the centre of the coil on the axis, the magnetic induction will become \( \frac{1}{\sqrt{8}} \) of its value at the centre of the coil?

• A. \( \frac{2R}{\sqrt{3}} \)
• B. \( R \sqrt{3} \)
• C. \( \frac{R}{2 \sqrt{3}} \)
• D. \( \frac{R}{\sqrt{3}} \)

Hint:

The magnetic induction on the axis of a coil decreases as the distance from the center increases, and is given by the formula that depends on both the radius and the distance.

Answer 1

The Correct Option is B

Step 1: Understanding the magnetic induction formula.
The magnetic induction \( B \) at a point on the axis of a circular current-carrying coil is given by: \[ B = \frac{\mu_0 I R^2}{2 (R^2 + x^2)^{3/2}} \] where \( I \) is the current, \( R \) is the radius of the coil, and \( x \) is the distance from the center of the coil along the axis.
Step 2: Using the given condition.
We are told that the magnetic induction at the point is \( \frac{1}{\sqrt{8}} \) of its value at the center. At the center, \( x = 0 \), so the magnetic induction is \( B_{\text{center}} = \frac{\mu_0 I}{2 R} \). Set the given condition: \[ \frac{1}{\sqrt{8}} \times \frac{\mu_0 I}{2 R} = \frac{\mu_0 I R^2}{2 (R^2 + x^2)^{3/2}} \] Solving this equation, we get: \[ x = R \sqrt{3} \] Step 3: Conclusion.
The correct answer is (B), \( R \sqrt{3} \).