Consider a rectangular loop of wire with dimensions a and b. One side of which is parallel to another long straight wire carrying current I₀, at a distance $\ell$, (as shown in the figure). The resistance of the loop is R and inductance is negligible.

Induced EMF in Rectangular Loop

Rectangular Loop Near a Straight Wire

Setup

• Long straight wire (current: I₀)
• Rectangular loop (dimensions: a × b, resistance R, negligible inductance)
• Nearest side of loop at distance ℓ from wire

Magnetic Flux Through the Loop

Magnetic field at distance r from wire:

B = μ₀I₀ / (2πr)

Total magnetic flux through loop:

Φ_B = (μ₀ I₀ b)/(2π) · ln((ℓ + a)/ℓ)

Induced EMF (if I₀ changes with time)

Ε_induced = - (μ₀ b)/(2π) · ln((ℓ + a)/ℓ) · dI₀/dt

Induced Current in Loop: I_loop = Ε_induced / R