Mutual Inductance of Line and Circle
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 62
9-8-2016
1680
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
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