Mutual Inductance of Line and Circle

A circular wire of radius a is insulated from an infinitely long straight wire in a tangential direction (see Figure 1.1). Find the mutual inductance.

Figure 1.1

SOLUTION

Label the circular wire conductor 1 and the straight wire conductor 2 (see Figure 1.2). Recall that the mutual inductance of two conductors is given by

(1)

where I₂ is the current flowing in conductor 2 and Φ₁₂ is the magnetic flux from conductor 2 through the surface bounded by conductor 1 (In modified

Figure 1.2

Gaussian units, M₁₂ = cΦ₁₂/I₂.

(2)

where B₂ is the magnetic field produced by the straight wire and dS is the element of area of the loop. The magnitude of the field due to a current in an infinite wire

(3)

is perpendicular to the plane of the loop. So, from (1)–(3), we have

(4)

Let x = 0, y = 0 be the center of the circle, and locate the infinite wire at x = -a. Then (4) becomes

(5)

Using the substitution u = cos 2θ, we obtain