1 FLORIDA ATLANTIC UNIVERSITY Department of Physics PHY 6346 Homework Assignments (Jackson: – Jackson’s homework problems) (A – Additional homework problems) Chapter 0 A0.1 Prove the following vector product rules: (a) ) ( ) ( ) ( B A C A C B C B A (b) C B A B C A C B A ) ( ) ( ) ( (c) ) )( ( ) )( ( ) ( ) ( C B D A D B C A D C B A (d) ) )( ( )) ( ( )) ( ( D C B A D C A B D C B A (e) Show that the volume of a triclinic primitive unit cell ( , a b c ) is: ˆ ( ) ( sin ) sin z V c a b c z ab abc 2 2 2 2 2 2 (cos cos cos ) sin 1 cos sin sin (cos cos cos ) sin V abc abc (f) Show that the volume of a rhombohedral (trigonal) primitive unit cell ( , ) a b c is: 3 4 2 2 3 ( ) sin cos (1 cos ) (1 cos )1 2 cos V c a b a a A0.2 Prove the following product rules: (a) f ( A ) ( ) ( ) f A A f (b) ) ( ) ( ) ( B A A B B A (c) f ( ) ( ) ( ) f A A f A A0.3 Prove the following second derivatives: (a) 0 ) ( A (b) 0 ) ( f (c) A A A 2 ) ( ) ( (d) g f g f f g fg 2 2 2 2 ) ( A0.4 If r is the coordinate of a point with respect to some origin, with magnitude r r , r r r / ˆ is a unit radial vector, and f(r) is a well-behaved function of r, show that (a) ˆ r r , (b) 2 ˆ 1 r r r , (c) 3 r , (d) 2 ˆ r r , (e) 0 r , (f) 2 ˆ 0 r r ,
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FLORIDA ATLANTIC UNIVERSITY Department of Physics
PHY 6346 Homework Assignments
(Jackson: – Jackson’s homework problems)
(A – Additional homework problems)
Chapter 0 A0.1 Prove the following vector product rules:
(a) )()()( BACACBCBA
(b) CBABCACBA
)()()(
(c) ))(())(()()( CBDADBCADCBA
(d) ))(())(())(( DCBADCABDCBA
(e) Show that the volume of a triclinic primitive unit cell ( ,a b c ) is:
ˆ( ) ( sin ) sinzV c a b c z ab abc
22 2 2 2
2
(cos cos cos )sin 1 cos sin sin (cos cos cos )
sinV abc abc
(f) Show that the volume of a rhombohedral (trigonal) primitive unit cell
( , )a b c is:
3 4 2 2 3( ) sin cos (1 cos ) (1 cos ) 1 2cosV c a b a a
A0.2 Prove the following product rules:
(a) f( A
)()() fAAf
(b) )()()( BAABBA
(c) f( )()() fAAfA
A0.3 Prove the following second derivatives:
(a) 0)( A
(b) 0)( f
(c) AAA
2)()(
(d) gfgffgfg 222 2)(
A0.4 If r
is the coordinate of a point with respect to some origin, with magnitude rr
,
rrr /ˆ
is a unit radial vector, and f(r) is a well-behaved function of r, show that
(a) ˆr r , (b) 2
ˆ1 r
r r , (c) 3r ,
(d) 2
rr
, (e) 0r , (f) 2
ˆ0
r
r ,
2
(g) )(4ˆ 3
2r
r
r
, (h) )(4
1 32 rrrr
,
(i) ˆ( )df
f r rdr
, (j) ˆ( ( )) 2 / /r f r f r f r ,
(k) ˆ( ( )) 0r f r , (l) ( )
ˆ ˆ ˆ ˆ ˆ( ) ( ) [ ( )] ( )f r f
a rf r a r a r r a rr r
.
(m) ( ) ( ) ( )r a a r a i L a , where ( )L i r is the angular momentum
operator.
(n) Given r r= = r r show that in spherical coordinates
2 2 2 2 2 22 2 cos 2 [cos cos sin sin cos( )]r r r r r r rr r r rr r
and cos cos cos sin sin cos( )
(o) Given r r= = r r show that in cylindrical coordinates
2 2 22 cos( ) ( )r r s s ss z z r
(p) Prove: 2 3
ˆ 1ˆ ˆ[ 3( ) ]
p rp p r r
r r
, where ˆ ˆˆrp p r p p is a constant
vector, and 0r .
(q) Prove: 2 3
ˆ 1ˆ ˆ[3( ) ]
m rm r r m
r r
, where ˆ ˆˆ
rm m r m m is a constant
vector, and 0r .
A0.5 Check the theorem of gradients using 32 24 yzxyxT , the points )0,0,0(a
,
)1,1,1(b
, and the three paths in Fig.1:
(a) (0,0,0) (1,0,0) (1,1,1);
(b) (0,0,0) (0,0,1) (1,1,1);
(c) (the parabolic path 2xz ; y = x.
A0.6 Check the gradient theorem for function )cossin(cos rT using the path shown
in Fig. 2.
Figure 3
3
A0.7 Check the divergence theorem for the function cos2r
r cos2r
sincos2r , using as your volume one octant of the sphere of radius R (Fig. 3).
A0.8 Check the Stokes’ theorem using the function ybxxay ˆ)(ˆ)(
( a and b are constants)
and the circular path of radius R, centered at the origin in the xy plane.
A0.9 Check Stokes’ theorem for the function zyˆ
, using the triangular surface shown in
Fig. 4.
A0.10 Check the divergence theorem for the function sin2r
r cos4 2r tan2r
, using the volume of the “ice-cream cone” shown in Fig. 5 (the top surface is spherical,
with radius R and centered at the origin).
A0.11 Prove the following integral formulas:
(a) SV S
addand
ˆ ;
(b) V S
daBndB
ˆ ;
(c) ˆ
S
n da dl
(d) ˆ( )n da dS
(e) Green’s first identity: V S
dand ˆ)( 2
(f) Green’s theorem: 2 2 ˆ( ) ( )V S
d nda
dann
S)(
(g) ( ) 0fJ g gJ f fg J d
Where ( )J r is localized but not necessarily divergenceless, ( )f r and ( )g r
are well-behaved functions of r .
(h) The integral S
a da is sometimes called the vector area of the surface S. If S
happens to be flat, then a is the ordinary (scalar) area, obviously.
Find the vector area of a hemispherical bowl of radius R.
Show that 0a for any closed surface. [Hint: use integral in A0.11(a)].
Figure 4
x
y
z
(0,2a,0)
(0,0,a)
(a,0,0)
300
z
y
x
R
Figure 5
4
Show that a is the same for all surface sharing the same boundary.
Show that 1
2a r d , where the integral is around the boundary line.
[Hint: One way to do it is to draw the cone subtended by the loop at the origin.
Divide the conical surface up into infinitesimal triangular wedges, each with
vertex at the origin and opposite side d , and exploit the geometrical interception
of the cross product.
(i) Show that ( ) ,c r d a c where, S
a da is the vector area of the loop surface,
r is the position vector of a point on the loop with respect to some origin, c is any
constant vector.
[Hint: Use integral ˆ
S
n da dl in A0.11(c), and product rule of ( )A B ].
A0.12 (a) Show that )(1
)( xk
kx ; (b) Show that )()( xx ;
(c) Show that )()( xx ; (d) Show that )()( xxx ;
(e) Show that )()/( 2 ctrccrt ;
(f) Show that )()/( 3 ctrccrt ;
(g) Show that
)()()( afdxaxxf
(h) Let )(x be the step function:
00
01)(
x
xx , show that )(x
dx
d
A0.13 Evaluate the following integrals:
(a) dxxxx )3()123(6
2
2
(b) 5
0)(cos dxxx
(c) dxxx )1(3
0
3
(d) dxxx
)2()3ln(
(e)
2
2)3()32( dxxx
(f) dxxxx )1()23(2
0
3
(g) dxxx )13(91
1
2
(h)
adxbx )(
A0.14 Evaluate the following integrals:
(a) daraarrallspace
)()( 22 , where a
is a fixed vector and a is its magnitude.
(b) 2
3(5 )V
r b r d , Where V is a cube of side 2, centered on the origin, and
zyb ˆ3ˆ4
.
(c) dcrccrrrV
)())(( 424 , where V is a sphere of radius 6 about the origin,
zyxc ˆ2ˆ3ˆ5
, and c is its magnitude.
5
(d) drerdrV
)()( 3 , where )3,2,1(d
, )1,2,3(e
, and V is a sphere of radius
1.5 centered at (2,2,2).
A0.15 Write down the expressions of )(3 rr
in Cartesian, Spherical and cylindrical
coordinates.
A0.16 Using Dirac delta functions and theta functions in the appropriate coordinates, express
the following charge distributions as three-dimensional charge densities )(r
:
(a) In spherical coordinates, a charge Q is uniformly distributed over a spherical shell of
radius R.
(b) In cylindrical coordinates, a charge per unit length is uniformly distributed over a
cylindrical surface of radius b.
(c) In cylindrical coordinates, a charge Q spreads uniformly over a flat circular disc of
radius R and negligible thickness.
(d) The same as part (c), but using spherical coordinates.
(e) In spherical coordinates, a charge Q is uniformly distributed over a circular ring of
radius a.
(f) In spherical coordinates, a charge Q is uniformly
distributed over a half of a circular ring with radius
R as shown in Fig. 6.
(g) In cylindrical coordinates, a charge Q is uniformly distributed over a half of a circular
ring with radius R as shown in Fig. 6.
(h) In cylindrical coordinates, a charge per unit length is uniformly distributed over
an infinite long straight wire.
(i) In spherical coordinates, a charge Q is uniformly distributed over a straight wire of
length 2b.
(j) In Cartesian coordinates, a charge per unit length is uniformly distributed over a
straight wire of length l which lies on the x-axis from lx to x=0.
(k) In Cartesian coordinates, a charge per unit length is uniformly distributed over a
straight wire of length l which lies on the x-axis from 2/lx to x= 2/l .
(l) In spherical coordinates, point charges are located as shown in the Fig. 7 (a) and (b).
(m) In spherical coordinates, a charge Q is uniformly distributed over a volume of
the “ice-cream cone” shown in Fig. 5
(n) In spherical coordinates, a charge Q is uniformly distributed over the spherical
surface of the “ice-cream cone” shown in Fig. 5
(o) In spherical coordinates, a charge Q is uniformly distributed over 1/8 of a
spherical shell as shown in Fig. 3.
x
y
z
-q
q
-q q
a a a
a
(a)
z
x
y
a
a
q
q
-2q
(b) Fig. 7
R y
x
z
Fig. 6
6
(p) In spherical coordinates, a charge Q is uniformly distributed over the volume of
1/8 of a sphere as shown in Fig. 3.
(q) In spherical coordinates, a line charge of length 2d with a total charge Q has a
linear charge density varying as ( 2 2d z ), where z is the distance from the
middle point.
(r) In spherical coordinates, a charge density 2( ) sin ( / 2) ( is a constant)
is glued over the surface of a spherical shell of radius R.
A0.17 Express the volume current density of a current loop of radius R, lying in the x-y
plane, centered at the origin and carrying a current I, using Dirac delta functions in
(a) spherical coordinates, and (b) cylindrical coordinates.