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MCQEasyJEE 2023Biot–Savart Law

JEE Physics 2023 Question with Solution

A circular loop of radius rr is carrying current I AI \, \text{A}. The ratio of magnetic field at the center of the circular loop and at a distance rr from the center of the loop on its axis is:

  • A

    1:321:3\sqrt{2}

  • B

    32:23\sqrt{2}:2

  • C

    22:12\sqrt{2}:1

  • D

    1:21:\sqrt{2}

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: A circular loop of radius rr carries current II.

Find: The ratio of magnetic field at the center and at a point on the axis at distance rr from the center.

The magnetic field on the axis of a current-carrying circular loop is given by

Baxis=μ0Ir22(r2+x2)3/2B_{\text{axis}} = \frac{\mu_0 I r^2}{2(r^2 + x^2)^{3/2}}

At the center of the loop, x=0x = 0, so

Bcenter=μ0I2rB_{\text{center}} = \frac{\mu_0 I}{2r}

At a distance rr on the axis, x=rx = r, so

Baxis=μ0Ir22(r2+r2)3/2=μ0Ir22(2r2)3/2=μ0I42rB_{\text{axis}} = \frac{\mu_0 I r^2}{2(r^2 + r^2)^{3/2}} = \frac{\mu_0 I r^2}{2(2r^2)^{3/2}} = \frac{\mu_0 I}{4\sqrt{2}r}

Therefore,

BcenterBaxis=μ0I/(2r)μ0I/(42r)=22\frac{B_{\text{center}}}{B_{\text{axis}}} = \frac{\mu_0 I/(2r)}{\mu_0 I/(4\sqrt{2}r)} = 2\sqrt{2}

So the ratio is 22:12\sqrt{2}:1.

The solution working gives 22:12\sqrt{2}:1, which matches option C. The solution label says B, but that label is inconsistent with the working. Therefore, the correct option is C.

Common mistakes

  • Using the magnetic field formula for the center at every point on the axis is wrong because the axial field depends on distance xx. Use Baxis=μ0Ir22(r2+x2)3/2B_{\text{axis}} = \frac{\mu_0 I r^2}{2(r^2+x^2)^{3/2}} for points away from the center.

  • Substituting x=rx=r incorrectly into (r2+x2)3/2(r^2+x^2)^{3/2} is a common error. At x=rx=r, it becomes (2r2)3/2=22r3(2r^2)^{3/2} = 2\sqrt{2}r^3, not 2r32r^3.

  • Matching the final numerical ratio to the wrong option can happen if only the printed option label is trusted. Here the working gives 22:12\sqrt{2}:1, so the defensible choice is the option containing that ratio.

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