Begin2.DVI

and take the curl of the second and fourth of the Maxwell’s equations to obtain

∇× (∇× E
) = ∇(∇·E
)−∇^2 E
=∇×

(

−∂

B


∂t

)

=−∂t∂

(

∇× B

)

=−μ 0

0 ∂

(^2) E

∂t^2
∇× (∇× B
) = ∇(∇·B
)−∇^2 B
=∇×
(
μ 0
0 ∂
E

∂t
)
=μ 0
0 ∂
∂t
(
∇× E

)
=−μ 0
0 ∂
(^2) B

∂t^2

The first and third of the Maxwell equations require that ∇· E
= 0 and ∇·B
= 0 so

that the vector fields E
and B
must satisfy the wave equations

∇^2 E
 =^1

c^2

∂^2 E


∂t^2

and ∇^2 B
=^1

c^2

∂^2 B


∂t^2

Here the product μ 0

0 =^1

c^2

, where c= 3 ×(10)^10 cm/sec is the speed of light.

Exercises

9-1. Solve each of the one-dimensional Laplace equations

d^2 U

dx^2 =0 , U =U(x) rectangular

d^2 U

dr^2 +

1

r

dU

dr =

1

r

d

dr

(

rdUdr

)

=0 , U =U(r) polar

d^2 U

dρ^2

+

2

ρ

dU

dρ

=

1

ρ^2

d

dρ

(

ρ^2

dU

dρ

)

=0 , U =U(ρ) spherical

(9 .162)

9-2. Verify that the velocity field V
=V 0 cos αˆe 1 −V 0 sin αˆe 2 , V 0 , α are constants

is both irrotational and solenoidal. Find and sketch the velocity field, streamlines.

Find the velocity potential. Note that for α= 0 the flow is a parallel flow and for

α=π 2 the flow is a vertical flow.

9-3. Verify that the velocity field V
= 2xˆe 1 − 2 yˆe 2 is both irrotational and

solenoidal. Find and sketch the vector field and the streamlines for 0 < x < 2 ,

0 < y < 2. Also find the velocity potential. The velocity field for this type of fluid

motion can be used to describe the flow in the vicinity of a corner.

9-4. For the velocity field
V = 2yˆe 1 + 2 xˆe 2 find and sketch the vector field and

streamlines. Find the velocity potential.