What is the magnetic field at the center of a circular loop of radius R carrying current I?

What is the magnetic field at the center of a circular loop of radius R carrying current I?

• A. B_center = μ0 I /(R)
• B. B_center = μ0 I /(2R)
• C. B_center = μ0 I /(√2 R)
• D. B_center = μ0 I /(4R)

Answer: (B)

The magnetic field at the center comes from summing the contributions of all current elements around the loop via the Biot-Savart law. For a circular loop, each element dℓ is at distance R from the center, and the angle between the element and the line to the center is 90°, so the differential field dB has magnitude μ0 I dℓ /(4π R^2). Adding up all these contributions around the full circle gives B = (μ0 I /(4π R^2)) ∮ dℓ = (μ0 I /(4π R^2)) (2πR) = μ0 I /(2R). The field points perpendicular to the loop, determined by the right-hand rule. This shows why the field varies with current and inversely with radius, and explains the 1/(2R) result rather than 1/R, 1/(√2 R), or 1/(4R).

The magnetic field at the center comes from summing the contributions of all current elements around the loop via the Biot-Savart law. For a circular loop, each element dℓ is at distance R from the center, and the angle between the element and the line to the center is 90°, so the differential field dB has magnitude μ0 I dℓ /(4π R^2). Adding up all these contributions around the full circle gives B = (μ0 I /(4π R^2)) ∮ dℓ = (μ0 I /(4π R^2)) (2πR) = μ0 I /(2R). The field points perpendicular to the loop, determined by the right-hand rule. This shows why the field varies with current and inversely with radius, and explains the 1/(2R) result rather than 1/R, 1/(√2 R), or 1/(4R).