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. μ0 I / (2R)
• B. μ0 I / R
• C. μ0 I / (4R)
• D. μ0 I / (2πR)

Answer: (A)

A current loop creates a magnetic field at its center that is the sum of the contributions from each tiny segment of wire. Using Biot-Savart, a small piece dl on the loop at distance R from the center gives a field dB with magnitude μ0 I dl /(4π R^2) and direction perpendicular to the plane of the loop. Since dl is tangential and r (the radius to the center) is radial, the angle between dl and r is 90°, so sinθ = 1 and all these contributions add along the same axis. Integrating around the loop, B = μ0 I /(4π R^2) ∮ dl. The total length around the loop is 2πR, so B = μ0 I /(4π R^2) × 2πR = μ0 I /(2R). The field through the center points perpendicular to the loop in the direction given by the right-hand rule. Therefore, the magnetic field at the center is μ0 I /(2R).

A current loop creates a magnetic field at its center that is the sum of the contributions from each tiny segment of wire. Using Biot-Savart, a small piece dl on the loop at distance R from the center gives a field dB with magnitude μ0 I dl /(4π R^2) and direction perpendicular to the plane of the loop. Since dl is tangential and r (the radius to the center) is radial, the angle between dl and r is 90°, so sinθ = 1 and all these contributions add along the same axis. Integrating around the loop, B = μ0 I /(4π R^2) ∮ dl. The total length around the loop is 2πR, so B = μ0 I /(4π R^2) × 2πR = μ0 I /(2R). The field through the center points perpendicular to the loop in the direction given by the right-hand rule. Therefore, the magnetic field at the center is μ0 I /(2R).