Begin2.DVI

I9-9. Integrate the equations ∂φ∂x=x 2 −+kyy 2 and∂φ∂y=x 2 kx+y 2 to obtain

φ=−ktan−^1

(

x

y

)

+C 1 and φ=ktan−^1

(y

x

)

+C 2

Show thattan−^1 h+ tan−^1 (^1

h

) =π

2

forh > 0 and then letc 1 =π 2 k, c 2 = 0to obtain the

potential

φ=k

[

π

2 −tan

− 1

(

x

y

)]

=ktan−^1

(y

x

)

Streamlines are circlesψ=x^2 +y^2 =c

I9-10.

(d) 2 xy x^2 −y^2

(e)x^2 +y^22 xy

I9-11. Show that

∂φ

∂x=

∂ψ

∂y =⇒

(

∂φ

∂r−

1

r

∂ψ

∂θ

)

cosθ=

(

∂ψ

∂r+

1

r

∂φ

∂θ

)

sinθ

and

∂φ

∂y=−

∂ψ

∂x =⇒

(

∂φ

∂r−

1

r

∂ψ

∂θ

)

sinθ=−

(

∂ψ

∂r+

1

r

∂ψ

∂θ

)

cosθ

If the above equations are to hold for all values ofθ, then the coefficient of thesinθ

andcosθterms must equal zero.

I9-12.

On upper semi-circle

∫

C 1 F~·d~r=−π/^4

On the lower semi-circle

∫

C 2 F~·d~r=π/^4

Solutions Chapter 9