9-36. Express the Maxwell equations (9.161) as a system of partial differential
equations.
9-37. Assume solutions to the Maxwell equations (9.161) are waves moving in
the x-direction only. This is accomplished by assuming exponential type solutions
having the form ei(kx−ωt)where iis an imaginary unit satisfying i^2 =− 1.
(a) Show that E =E(x, t) = E 0 ei(kx−ωt) and B =B(x, t) = B 0 ei(kx−ωt)are solutions
of Maxwell’s equations in this special case.
(b) Show that B 0 =
k
ω(
ˆe 1 ×E 0 )
(c) Show that the waves for E and B are mutually perpendicular.
9-38. Consider the following vector fields:
B a magnetic field intensity with units of amp /m
E an electrostatic intensity vector with units of volts/m
Q a heat flow vector with units of joules/cm^2 ·sec
V a velocity vector with units of cm /sec
(a) Assign units of measurement to the following integrals and interpret the mean-
ings of these integrals:
(a)
∫∫
S
E·dS (b)
∫∫
S
Q·dS (c)
∫∫
S
V ·dS (d)
∫
C
B·dr
(b) Assign units of measurements to the quantities:
(a) curl H (b) div E (c) div Q (d) div V
9-39. Solve each the following vector differential equations
(a)
dy
dt =ˆe^1 t+ˆe^3 sin t (b)
d^2 y
dt^2 =ˆe^1 sin t+ˆe^2 cos t (c)
dy
dt = 3 y+ 6 ˆe^3
9-40. Solve the simultaneous vector differential equations ddty^1 =y 2 , ddty^2 =−y 1
9-41. A particle moves along the spiral r =r(θ) = r 0 eθcotα, where r 0 and α are
constants. If θ=θ(t)is such that dθ
dt
=ω=constant, find the components of velocity
in the direction r and in the direction perpendicular to r.
