Question

The magnetic field at the centre of a current carrying circular loop of radius \(R\) is \(16\,\mu\text{T}\). The magnetic field at a distance \(x=\sqrt{3}R\) on its axis from the centre is ____ \(\mu\text{T}\).

• A. 4
• B. 8
• C. \(2\sqrt{2}\)
• D. 2

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The magnetic field produced by a circular loop depends on the distance from the center along the axis according to Biot-Savart Law applications.
Step 2: Key Formula or Approach:
Field at center: \(B_c = \frac{\mu_0 I}{2R}\).
Field on axis at distance \(x\): \(B_x = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}\).
Step 3: Detailed Explanation:
We can write the axial field in terms of the central field:
\[ B_x = B_c \cdot \frac{R^3}{(R^2 + x^2)^{3/2}} \] Given \(x = \sqrt{3}R\):
\[ B_x = 16 \cdot \frac{R^3}{(R^2 + 3R^2)^{3/2}} \] \[ B_x = 16 \cdot \frac{R^3}{(4R^2)^{3/2}} = 16 \cdot \frac{R^3}{8R^3} \] \[ B_x = \frac{16}{8} = 2\ \mu\text{T} \] Step 4: Final Answer:
The magnetic field at the given axial point is \(2\ \mu\text{T}\).