622
13 Electromagnetism I
where the speed v = −dx/dt for the speed with which the connecting
rod is pulled out of the magnetic field. The emf Blv sets up a current in
the loop given by
i =
ξ
R
=
Blv
R
(3)
(b) The total power delivered to the resistor is just the Joule heat given by
P I = i
2 R =
B 2 l 2 v 2
R
Note that energy conservation tells us that for steady motion of the rod
the external agent must provide power equal to the Joule heat. That this
is so is borne out from
P = F 3 v =
B 2 l 2 v 2
R
·
(c) The current the loop produces forces F 1 , F 2 and F 3 on the three sides of
the loop in accordance with
F = il × B
(4)
Because F 1 and F 2 are equal and opposite, they cancel each other, while
F 3 which opposes the motion of the sliding rod is given by (4) and (3) in
magnitude as
F 1 = ilB sin 90
◦
=
B 2 l 2 v
R
(5)
13.3.4 Magnetic Energy, Magnetic Dipole Moment
13.60 Elementary magnetic moment
dμ = (di)(dA) =
ω qrdr
π R 2
(π r
2
) =
ω qr 3 dr
R 2
where dA is the area enclosed by r and the value of di is used from (2) of
prob. (13.37).
Therefore the magnetic moment of the disc is