Download 習題九 29.17. A clockwise current through the loop produces a

習題九
29.17. A clockwise current through the loop produces a magnetic field at the center of the loop that points
down into the plane of the page. So, by Lenz’s law, to induce a clockwise current in the loop, we must have
a changing magnetic field in the loop that either points down into the page with its magnitude decreasing, or
one that points up out of the page with its magnitude increasing. When the current is in the direction
indicated in Figure P29.17, it produces a magnetic field that points down into the page at the location of the
loop. If we take this as the positive current direction, then the field will be into the page with its magnitude
decreasing when the current’s phase is between one-half pi and pi, that is, when
(2n  12 ) /  t  (2n  1) /, for integer n 
1
2
  t  2n   or
, 2, 1,0,1,2, . The field will be out of the page with its
magnitude increasing when the current’s phase is between pi and three-halves pi, that is, when
(2n  1) /  t  (2n  32 ) /. So, the induced current will be clockwise when the phase of the current in
the wire is between one-half pi and three-halves pi, that is, when (2n  12 ) /  t  (2n  32 ) /.
29.42. (a) Following Principles Example 29.7, we can relate the magnitude of the electric field to the rate
of change of the magnetic field. Because of the cylindrical symmetry of the magnetic field, the electric field
lines must form circles around the center of the magnetic field. Evaluating the line integral of the electric
field around such a circular path and the rate of change of magnetic flux through the area enclosed by the
path in Eq. 29.17, we have
dB
dt
dB
2 rE   r 2
dt
 Ed

E  12 r
dB 1 d
 2r
Bmax sin t  12 rBmax cos t
dt
dt
(b) In the case where r  R , we have to change the expression for the rate of change of magnetic flux
through the area enclosed by the circular path. We have instead
2 rE   R 2
E
dB
dt
R 2 dB R 2 d
R2

Bmax sin t 
Bmax cos t
2r dt
2r dt
2r
(c) At t  0, the magnetic field is zero and increasing in the positive z direction, so by Lenz’s law the
induced magnetic field is in the negative z direction, and the electric field lines encircle the z axis clockwise
as seen from the positive z axis.
29.43. Following Principles Example 29.7, we can relate the magnitude of the electric field to the rate of
change of the magnetic field. Because of the cylindrical symmetry of the magnetic field, the electric field
lines must form circles around the center of the magnetic field. Evaluating the line integral of the electric
field around such a circular path and the rate of change of magnetic flux through the area enclosed by the
path in Eq. 29.17, we have
dB
dt
dB
2 rE   r 2
dt
 Ed

dB 2 E 2(3Crt 2 )


 6Ct 2
dt
r
r
which we can integrate to find the magnitude of the magnetic field. Because our expression for the rate of
change of the magnetic field increases steadily from zero, and B  0 at t  0, we have
B
t
0
t
dB
dt   6Ct 2 dt  2Ct 3
0
dt