Mutual Inductance between a Long, Straight Wire and Rectangular Loop Laying in Same Plane

Mutual Inductance between a Long, Straight Wire and Rectangular Loop Laying in Same Plane

• Consider a rectangular loop with sides a and b as shown in the Fig. 8.11.1. A straight, long conductor is kept parallel to the longer side of the loop, along z-axis. A loop is kept at distance d from long conductor as shown.

• Consider that long, straight wire is circuit 1, while a rectangular loop is circuit 2. The magnetic field intensity at a distance ρ from the long conductor, due to current I₁ in it can be expressed using Ampere's circuital law.

• The flux linkage in the circuit 2 i.e. a rectangular loop due to current II, in circuit 1 i.e. straight long conductor is denoted as λ₁₂. The flux linkage in rectangular loop is given by,

• Solving above equation, we get,

• Assuming single turn circuit, the mutual inductance is given by,

• If the loop in parallel with a long, straight conductor is square loop with each side of 'a' units, then the mutual inductance between a square loop and a long straight conductor is given by

Ex. 8.11.1 A straight long wire is situated parallel to one side of a square coil. Each side of the coil has a length of 10 cm. The distance between straight wire and the centre of the coil is 20 cm. Find mutual inductance of the system. Derive formula used.

Sol. : Given : a = 10 cm = 10 × 10^-2 = 0.1 m d = 20 cm = 20 × 10^-2 = 0.2 m

The mutual inductance between the long straight wire and square coil is given by,

Review Question

1. Determine mutual inductance between straight long wire and a square loop.