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Formulas/physics/Moving Charges/Field at Centre of Circular Loop

Field at Centre of Circular Loop

Special case of the arc formula for θ = 2π. For N turns: B = μ₀NI/2R. Direction along the axis given by the right-hand curl rule.
Class 12
Derivation

From the Arc Formula

From mc11, the field at the centre of a circular arc of radius RR subtending angle θ\theta:

B=μ0Iθ4πRB = \frac{\mu_0 I\theta}{4\pi R}

For a complete circular loop, θ=2π\theta = 2\pi:

B=μ0I⋅2π4πRB = \frac{\mu_0 I \cdot 2\pi}{4\pi R} B=μ0I2R\boxed{B = \frac{\mu_0 I}{2R}}

For N Turns

B=μ0NI2RB = \frac{\mu_0 N I}{2R}

Each turn contributes independently; superposition gives NN times the single-turn result.

Direction

Right-hand curl rule: curl the fingers in the direction of current flow around the loop — the thumb points in the direction of B⃗\vec{B} at the centre.

Key Values

ConfigurationField at centre
Full loop, 1 turnμ0I/2R\mu_0 I / 2R
Semicircleμ0I/4R\mu_0 I / 4R
Quarter circleμ0I/8R\mu_0 I / 8R
Note
This result applies only at the geometric centre of the loop. The field on the axis (any other point) requires the full axial field formula from mc13.