How does the magnetic field strength at the center of a circular loop change if the radius of the loop is doub

Question

How does the magnetic field strength at the center of a circular loop change if the radius of the loop is doubled while keeping the current constant?

Examoogle · Accepted Answer

The correct answer is Option B: It becomes half. The magnetic field strength at the center of a circular current-carrying loop is inversely proportional to its radius. Therefore, when the radius of the loop is doubled while keeping the current constant, the magnetic field strength becomes half of its initial value.

Step-by-Step Solution & Concepts:
The magnetic field strength at the center of a circular current-carrying loop is inversely proportional to its radius. Therefore, when the radius of the loop is doubled while keeping the current constant, the magnetic field strength becomes half of its initial value.

Working Principle: According to the Biot-Savart law applied to a circular loop, the magnetic field $B$ at the center is given by the formula $B = \frac{\mu_0 I}{2R}$. Since the magnetic field is inversely proportional to the radius $R$ ($B \propto \frac{1}{R}$), doubling the radius ($R' = 2R$) results in the magnetic field being halved ($B' = \frac{B}{2}$), provided the current $I$ remains constant.

Key Formulas: $B = \frac{\mu_0 I}{2R}$ — Magnetic field at the center of a circular loop; $B \propto \frac{1}{R}$ — Inverse proportionality between magnetic field and radius

Key Points: The magnetic field at the center of a circular coil depends directly on the current flowing through it.; The magnetic field is inversely proportional to the radius of the circular loop.; Doubling the radius reduces the magnetic field strength by a factor of 2.

Applications: Design of galvanometers and electromagnetic coils; Magnetic resonance imaging (MRI) gradient coil considerations

Additional Info: $\mu_0$ represents the permeability of free space, having a standard value of $4\pi 	imes 10^{-7} 	ext{ T}\cdot	ext{m}/	ext{A}$.. Option A is incorrect because doubling the radius decreases the field, it does not double it.. Option C is incorrect because the field changes as the radius changes.. Option D is incorrect because a fourfold change would occur if the radius were halved or if squared relationships were involved.

Answer: B is correct — The magnetic field at the center of a circular loop is inversely proportional to its radius, so doubling the radius halves the magnetic field strength.

Exam Tip: Always identify whether variables are directly or inversely proportional in electromagnetic formulas to quickly solve ratio-based MCQ problems in competitive exams.

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Option A: It becomes double

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Option C: It remains the same

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Option D: It becomes four times

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