Question

A circular coil has radius ' \(R\) '. The distance from centre on axis where induction is \(1/27\) th of its value at centre is.

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

Hint:

Magnetic field on axis decreases rapidly as distance from the center increases.

Answer 1

The Correct Option is C

Step 1: Formula
\(B_{axis} = B_{centre} \cdot \left( \frac{R^2}{R^2 + x^2} \right)^{3/2}\).

Step 2: Calculation
\(\frac{1}{27} = \left( \frac{R^2}{R^2 + x^2} \right)^{3/2}\).
Taking cube root: \(\frac{1}{3} = \left( \frac{R^2}{R^2 + x^2} \right)^{1/2}\).
Squaring: \(\frac{1}{9} = \frac{R^2}{R^2 + x^2} \implies R^2 + x^2 = 9R^2\).
\(x^2 = 8R^2 \implies x = \sqrt{8}R = 2\sqrt{2}R\).
Final Answer: (C)