Ex=−
dV
dx
=−d
dx(
kQ
r)
=
kQ
r^2Page 594:
(a) The voltage (height) increases as you move to the east or north. If we let the positivex
direction be east, and choose positiveyto be north, then dV/dxand dV/dyare both positive.
This means thatExandEyare both negative, which makes sense, since the water is flowing in
the negativexandydirections (south and west).
(b) The electric fields are all pointing away from the higher ground. If this was an electrical
map, there would have to be a large concentration of charge all along the top of the ridge, and
especially at the mountain peak near the south end.
Page 606:
(a) The energy density depends onE·E, which equalsE^2 x+E^2 y+Ez^2.
(b) SinceExis squared, reversing its sign has no effect on the energy density. This makes sense,
because otherwise we’d be saying that the positive and negativexaxes in space were somehow
physically different in their behavior, which would violate the symmetry of space.
Page 606:
N−^1 m−^2 C^2 V^2 m−^2 m^2 m = N−^1 m−^1 C^2 V^2
= N−^1 m−^1 J^2
= J−^1 J^2
= JPage 615:
Yes. The mass has the same kinetic energy regardless of which direction it’s moving. Friction
coverts mechanical energy into heat at the same rate whether the mass is sliding to the right
or to the left. The spring has an equilibrium length, and energy can be stored in it either by
compressing it (x <0) or stretching it (x >0).
Page 616:
Velocity,v, is the rate of change of position,x, with respect to time. This is exactly analogous
toI= ∆q/∆t.
Page 626:
Say we’re looking foru=
√
z, i.e., we want a numberuthat, multiplied by itself, equalsz.
Multiplication multiplies the magnitudes, so the magnitude ofucan be found by taking the
square root of the magnitude ofz. Since multiplication also adds the arguments of the numbers,
squaring a number doubles its argument. Therefore we can simply divide the argument ofzby
two to find the argument ofu. This results in one of the square roots ofz. There is another one,
which is−u, since (−u)^2 is the same asu^2. This may seem a little odd: ifuwas chosen so that