Derive the equation relating the total charge QQ that flows through a search coil (Conceptual Example 29.3) to the magnetic-field magnitude BB. The search coil has NN turns, each with area AA, and the flux through the coil is decreased from its initial maximum value to zero in a time ΔtΔt. The resistance of the coil is RR, and the total charge is Q=IΔtQ=IΔt, where II is the average current induced by the change in flux.

Explanation:

flux through the coil = NBA

Change in flux = NBA - 0 = NBA

rate of change of flux = NBA / Δt

emf induced = NBA / Δt

current i = emf / resistance

= NBA / (RΔt)

Charge flowing through the search coil

=  NBA Δt/ (RΔt)

Q = NBA/R

see the picture for the vector

the magnitude of the vector by pythagorean theorem

r = √( (-10)^2 + (3)^2 ) = √109

magnitude = √109 units

the direction angle between vector and x-axis is

tan^(-1)(3/10) = 16.699

so the direction is approximately 16.7 degrees north of west