Answer to Question #168438 in Electricity and Magnetism for beth

Question

A conducting loop with area 0.15 m2and resistance 6.0 Ω lies in the x−y plane. A spatially uniform magnetic field points in the + z direction. The field varies with time according to B_z=at2−b, where a= 2.0 T/s2andb= 8.0 T. Find the loopcurrent magnitude and direction

(a) at t= 3.0 s and

(b) when B_z= 0 T.

Answer 1

$\Phi=SBcos \alpha,$

$\alpha=0,$

$\Phi=SB,$

$\varepsilon=-\frac{d\Phi}{dt}=-S\frac{dB}{dt},$

$\varepsilon=-S(2at)=-2atS,$

$I=\frac{\varepsilon}{R},$

a)

$I=-\frac{2atS}{R}=-\frac{2\cdot 2\cdot 3\cdot0.15}{6}=-0.3~A,$

b)

$B=0,$

$at^2=b,$

$t=\sqrt{\frac ba},$

$I=-\frac{2\sqrt{ab}S}{R},$

$I=-\frac{2\sqrt{2\cdot 8}\cdot 0.15}{6}=-0.2~A.$