Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.10 EXERCISES

11.24 Prove equation (11.22) and, by takingb=zx^2 i+zy^2 j+(x^2 −y^2 )k, show that the
two integrals

I=

∫

x^2 dV and J=

∫

cos^2 θsin^3 θcos^2 φdθdφ,

both taken over the unit sphere, must have the same value. Evaluate both directly
to show that the common value is 4π/15.
11.25 In a uniform conducting medium with unit relative permittivity, charge densityρ,
current densityJ, electric fieldEand magnetic fieldB, Maxwell’s electromagnetic
equations take the form (withμ 0  0 =c−^2 )

(i)∇·B= 0, (ii)∇·E=ρ/ 0 ,

(iii)∇×E+B ̇= 0 ,(iv)∇×B−(E ̇/c^2 )=μ 0 J.

The density of stored energy in the medium is given by^12 ( 0 E^2 +μ− 01 B^2 ). Show

that the rate of change of the total stored energy in a volumeVis equal to

−

∫

V

J·EdV−

1

μ 0

∮

S

(E×B)·dS,

whereSis the surface boundingV.
[ The first integral gives the ohmic heating loss, whilst the second gives the
electromagnetic energy flux out of the bounding surface. The vectorμ− 01 (E×B)
is known as the Poynting vector. ]
11.26 A vector fieldFis defined in cylindrical polar coordinatesρ, θ, zby

F=F 0

(

xcosλz

a

i+

ycosλz

a

j+(sinλz)k

)

≡

F 0 ρ

a

(cosλz)eρ+F 0 (sinλz)k,

wherei,jandkare the unit vectors along the Cartesian axes andeρis the unit

vector (x/ρ)i+(y/ρ)j.

(a) Calculate, as a surface integral, the flux ofFthrough the closed surface

bounded by the cylindersρ=aandρ=2aand the planesz=±aπ/2.

(b) Evaluate the same integral using the divergence theorem.

11.27 The vector fieldFis given by

F=(3x^2 yz+y^3 z+xe−x)i+(3xy^2 z+x^3 z+yex)j+(x^3 y+y^3 x+xy^2 z^2 )k.

Calculate (a) directly, and (b) by using Stokes’ theorem the value of the line

integral

∫

LF·dr,whereLis the (three-dimensional) closed contourOABCDEO
defined by the successive vertices (0, 0 ,0), (1, 0 ,0), (1, 0 ,1), (1, 1 ,1), (1, 1 ,0), (0, 1 ,0),
(0, 0 ,0).
11.28 A vector force fieldFis defined in Cartesian coordinates by

F=F 0

[(

y^3

3 a^3

+

y

a

exy /a

2

+1

)

i+

(

xy^2

a^3

+

x+y

a

exy /a

2

)

j+

z

a

exy /a

2

k

]

.

Use Stokes’ theorem to calculate

∮

L

F·dr,

whereLis the perimeter of the rectangleABCDgiven byA=(0, 1 ,0),B=(1, 1 ,0),

C=(1, 3 ,0) andD=(0, 3 ,0).