1 \begin{problem*}{31.46}
2 A circular loop of wire of resistance $R=0.500\U{\Ohm}$ and radius
3 $r=8.00\U{cm}$ is in a uniform magnetic field directed out of the page
4 as in Figure~P31.46. If a clockwise current of $I=2.50\U{mA}$ is
5 induced in the loop, \Part{a} is the magnetic field increasing or
6 decreasing in time? \Part{b} Find the rate at which the field is
16 Vector B = BField(phi=90);
17 vector_field(width=2.5*r, height=2.5*r, v=B);
18 draw(scale(r)*unitcircle);
19 Distance Dr = Distance((0,0), (r,0), "$r$"); Dr.draw();
20 draw(arc((0,0), r+dr, angle1=10, angle2=-10), CurrentPen, ArcArrow);
21 label("$I$", (r+dr, 0), align=E);
28 A clockwise current induces an inward magnetic field inside the loop,
29 so the external flux must be increasingly out of the page. Therefore,
30 the magnetic field is \ans{increasing}.
33 The induced \EMF\ must be
38 This is related to the changing field via magnetic flux.
40 |\EMF| &= |-\deriv{t}{\Phi_B}| = \deriv{t}{AB} = A\deriv{t}{B}
41 = \pi r^2 \deriv{t}{B} \\
42 \deriv{t}{B} &= \frac{\EMF}{\pi r^2} = \frac{IR}{\pi r^2}
43 = \ans{62.2\U{T/s}} \;.