Faraday's Law Need help with explanation

[1010101]

Homework Statement

http://em.groups.et.byu.net/embook/ch6/mod6.4.html

^^The link has the problem

Three wire-loop configurations are shown below. In all cases, the long straight wire is oriented along the z-direction and carried current I=Io cos wt. The loop in (a) lies in the y-z plane. In contrast, the loops in (b) and (c) lie in the x-y plane. All three loops are of equal area. Explain which loop configuration will induce a V emf with the largest amplitude.

Homework Equations

The Attempt at a Solution

I know the answer is (a), but can someone explain to me the reason configuration (A) induces the largest V emf?? compared to (b) or (c)??

 

[aralbrec]

The emf induced is proportional to the rate of change of flux penetrating the area of the loop.

ε = - dψ/dt
(which comes from ∇×E = -∂B/dt integrated over the area enclosed by the loop)

The current I sets up a magnetic field that circles the wire. Take your right hand, point your thumb in the direction of the current and your folded fingers show you how the magnetic field circles the wire.

For b and c, the field is parallel to the plane containing the wire loop so no flux passes through the area enclosed by the loop. ∂ψ/dt is zero so no emf is induced.

For a, the field penetrates the loop area at 90 degrees. Because the current is sinusoidal and the B it generates is proportional to the current, the B penetrating the loop is also sinusoidal, ie is changing. Therefore an emf is induced in the loop.