Question

State the formula for magnetic induction produced by a current in a circular arc of a wire. Hence find the magnetic induction at the centre of a current carrying circular looр.

Answer 1

Magnetic Induction Produced by a Current in a Circular Arc of a Wire:

The magnetic induction (magnetic field, B) at the center of a circular arc of a wire carrying a current I is given by:

B = `(μ_0Iθ)/(4R)`

Where,

• B = Magnetic induction at the center
• μ₀ = Permeability of free space (4π × 10⁻⁷ T·m/A)
• I = Current flowing through the wire (A)
• θ = Angle subtended by the arc at the center (in radians)
• R = Radius of the circular arc (m)

This formula is derived using Biot-Savart’s Law, which states that a small current element produces a magnetic field, and the total field is obtained by integrating over the arc length.

Magnetic Induction at the Center of a Current-Carrying Circular Loop:

For a full circular loop, the subtended angle is:

θ = 2π

Substituting θ = 2π into the formula for a circular arc:

B = `(μ_0I(2pi))/(4R)`

B = `(μ_0I)/(2R)`