Exercises

11.10 EXERCISES
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everywhere except on the axis ρ = 0, where v has a singularity. Therefore C v · dr
equals zero for any path C that does not enclose the vortex line on the axis and
2π if C does enclose the axis. In order for Stokes’ theorem to be valid for all
paths C, we therefore set
∇ × v = 2πδ(ρ),
where δ(ρ) is the Dirac delta function, to be discussed in subsection 13.1.3. Now,
since ∇ × v = 0, except on the axis ρ = 0, there exists a scalar potential ψ such
that v = ∇ψ. It may easily be shown that ψ = φ, the polar angle. Therefore, if C
does not enclose the axis then
0
0
v · dr = dφ = 0,
C
and if C does enclose the axis,
0
v · dr = ∆φ = 2πn,
C
where n is the number of times we traverse C. Thus φ is a multivalued potential.
Similar analyses are valid for other physical systems – for example, in magnetostatics we may replace the vortex lines by current-carrying wires and the velocity
field v by the magnetic field B.
11.10 Exercises
11.1
The vector field F is defined by
F = 2xzi + 2yz 2 j + (x2 + 2y 2 z − 1)k.
11.2
11.3
11.4
11.5
Calculate ∇ × F and deduce that F can be written F = ∇φ. Determine the form
of φ.
The vector field Q is defined by
Q = 3x2 (y + z) + y 3 + z 3 i + 3y 2 (z + x) + z 3 + x3 j + 3z 2 (x + y) + x3 + y 3 k.
Show that Q is a conservative
field, construct its potential function and hence
evaluate the integral J = Q · dr along any line connecting the point A at
(1, −1, 1) to B at (2, 1, 2).
F is a vector field xy 2 i + 2j + xk, and L is a path
by x = ct,
parameterised
y = c/t,
z = d for the range 1 ≤ t ≤ 2. Evaluate (a) L F dt, (b) L F dy and (c) L F · dr.
By making an appropriate choice for the functions P (x, y) and Q(x, y) that appear
in Green’s theorem in a plane, show that the integral of x − y over the upper half
of the unit circle centred on the origin has the value − 23 . Show the same result
by direct integration in Cartesian coordinates.
Determine the point of intersection P , in the first quadrant, of the two ellipses
y2
x2
y2
x2
+ 2 = 1 and
+ 2 = 1.
a2
b
b2
a
Taking b < a, consider the contour L that bounds the area in the first quadrant
that is common to the two ellipses. Show that the parts of L that lie along the
coordinate axes contribute nothing to the line integral around L of x dy − y dx.
Using a parameterisation of each ellipse similar to that employed in the example
409
LINE, SURFACE AND VOLUME INTEGRALS
11.6
in section 11.3, evaluate the two remaining line integrals and hence find the total
area common to the two ellipses.
By using parameterisations of the form x = a cosn θ and y = a sinn θ for suitable
values of n, find the area bounded by the curves
x2/5 + y 2/5 = a2/5
11.7
and x2/3 + y 2/3 = a2/3 .
Evaluate the line integral
0
I=
y(4x2 + y 2 ) dx + x(2x2 + 3y 2 ) dy
C
2
11.8
2
around the ellipse x /a + y 2 /b2 = 1.
Criticise the following ‘proof’ that π = 0.
(a) Apply Green’s theorem in a plane to the functions P (x, y) = tan−1 (y/x) and
Q(x, y) = tan−1 (x/y), taking the region R to be the unit circle centred on the
origin.
(b) The RHS of the equality so produced is
y−x
dx dy,
2
2
R x +y
which, either from symmetry considerations or by changing to plane polar
coordinates, can be shown to have zero value.
(c) In the LHS of the equality, set x = cos θ and y = sin θ, yielding P (θ) = θ
and Q(θ) = π/2 − θ. The line integral becomes
2π π
− θ cos θ − θ sin θ dθ,
2
0
which has the value 2π.
(d) Thus 2π = 0 and the stated result follows.
11.9
A single-turn coil C of arbitrary shape is placed in a magnetic field B and carries
a current I. Show that the couple acting upon the coil can be written as
B(r · dr).
M = I (B · r) dr − I
C
11.10
11.11
C
For a planar rectangular coil of sides 2a and 2b placed with its plane vertical
and at an angle φ to a uniform horizontal field B, show that M is, as expected,
4abBI cos φ k.
Find the vector area S of the part of the curved surface of the hyperboloid of
revolution
x2
y2 + z2
−
=1
2
a
b2
that lies in the region z ≥ 0 and a ≤ x ≤ λa.
An axially symmetric solid body with its axis AB vertical is immersed in an
incompressible fluid of density ρ0 . Use the following method to show that,
whatever the shape of the body, for ρ = ρ(z) in cylindrical polars the Archimedean
upthrust is, as expected, ρ0 gV , where V is the volume of the body.
Express the vertical component of the resultant force on the body, − p dS,
where p is the pressure, in terms of an integral; note that p = −ρ0 gz and that for
an annular surface element of width dl, n · nz dl = −dρ. Integrate by parts and
use the fact that ρ(zA ) = ρ(zB ) = 0.
410
11.10 EXERCISES
11.12
Show that the expression below is equal to the solid angle subtended by a
rectangular aperture, of sides 2a and 2b, at a point on the normal through its
centre, and at a distance c from the aperture:
b
ac
Ω=4
dy.
2
2
2
2
2 1/2
0 (y + c )(y + c + a )
By setting y = (a2 + c2 )1/2 tan φ, change this integral into the form
φ1
4ac cos φ
dφ,
c2 + a2 sin2 φ
0
where tan φ1 = b/(a2 + c2 )1/2 , and hence show that
ab
.
Ω = 4 tan−1
c(a2 + b2 + c2 )1/2
11.13
11.14
11.15
11.16
A vector field a is given by −zxr −3 i−zyr−3 j+(x2 +y 2 )r−3 k, where r2 = x2 +y 2 +z 2 .
Establish that the field is conservative (a) by showing that ∇ × a = 0, and (b) by
constructing its potential function φ.
2
2
A vector field a is given by (z 2 + 2xy) i + (x
+ 2yz) j + (y + 2zx) k. Show that
a is conservative and that the line integral a · dr along any line joining (1, 1, 1)
and (1, 2, 2) has the value 11.
A force F(r) acts on a particle at r. In which of the following cases can F be
represented in terms of a potential? Where it can, find the potential.
2
2(x − y)
r
(a) F = F0 i − j −
r
exp
− 2 ;
a2
a
2
(x2 + y 2 − a2 )
F0
r
zk +
(b) F =
r exp − 2 ;
2
a
a
a
a(r × k)
(c) F = F0 k +
.
2
r
One of Maxwell’s electromagnetic equations states that all magnetic fields B
are solenoidal (i.e. ∇ · B = 0). Determine whether each of the following vectors
could represent a real magnetic field; where it could, try to find a suitable vector
potential A, i.e. such that B = ∇ × A. (Hint: seek a vector potential that is parallel
to ∇ × B.):
B0 b
[(x − y)z i + (x − y)z j + (x2 − y 2 ) k] in Cartesians with r2 = x2 + y 2 + z 2 ;
r3
3
B0 b
[cos θ cos φ êr − sin θ cos φ êθ + sin 2θ sin φ êφ ] in spherical polars;
(b)
r3 1
zρ
in cylindrical polars.
+
ê
ê
(c) B0 b2
ρ
z
(b2 + z 2 )2
b2 + z 2
(a)
11.17
The vector field f has components yi−xj+k and γ is a curve given parametrically
by
r = (a − c + c cos θ)i + (b + c sin θ)j + c2 θk,
11.18
0 ≤ θ ≤ 2π.
Describe the shape of the path γ and show that the line integral γ f · dr vanishes.
Does this result imply that f is a conservative field?
A vector field a = f(r)r is spherically symmetric and everywhere directed away
from the origin. Show that a is irrotational, but that it is also solenoidal only if
f(r) is of the form Ar−3 .
411
LINE, SURFACE AND VOLUME INTEGRALS
11.19
Evaluate the surface integral r · dS, where r is the position vector, over that part
of the surface z = a2 − x2 − y 2 for which z ≥ 0, by each of the following methods.
(a) Parameterise the surface as x = a sin θ cos φ, y = a sin θ sin φ, z = a2 cos2 θ,
and show that
r · dS = a4 (2 sin3 θ cos θ + cos3 θ sin θ) dθ dφ.
(b) Apply the divergence theorem to the volume bounded by the surface and
the plane z = 0.
11.20
Obtain an expression for the value φP at a point P of a scalar function φ that
satisfies ∇2 φ = 0, in terms of its value and normal derivative on a surface S that
encloses it, by proceeding as follows.
(a) In Green’s second theorem, take ψ at any particular point Q as 1/r, where r
is the distance of Q from P . Show that ∇2 ψ = 0, except at r = 0.
(b) Apply the result to the doubly connected region bounded by S and a small
sphere Σ of radius δ centred on P.
(c) Apply the divergence theorem to show that the surface integral over Σ
involving 1/δ vanishes, and prove that the term involving 1/δ 2 has the value
4πφP .
(d) Conclude that
1
1
∂ 1
1 ∂φ
dS +
φ
φP = −
dS.
4π S ∂n r
4π S r ∂n
This important result shows that the value at a point P of a function φ
that satisfies ∇2 φ = 0 everywhere within a closed surface S that encloses P
may be expressed entirely in terms of its value and normal derivative on S.
This matter is taken up more generally in connection with Green’s functions
in chapter 21 and in connection with functions of a complex variable in
section 24.10.
11.21
Use result (11.21), together with an appropriately chosen scalar function φ, to
prove that the position vector r̄ of the centre of mass of an arbitrarily shaped
body of volume V and uniform density can be written
0
1
1 2
r̄ =
r dS.
V S 2
11.22
A rigid body of volume V and surface S rotates with angular velocity ω. Show that
0
1
u × dS,
ω=−
2V S
11.23
where u(x) is the velocity of the point x on the surface S.
Demonstrate the validity of the divergence theorem:
(a) by calculating the flux of the vector
αr
(r2 + a2 )3/2
√
through the spherical surface |r| = 3a;
(b) by showing that
3αa2
∇·F= 2
(r + a2 )5/2
and evaluating
the volume integral of ∇ · F over the interior of the sphere
√
|r| = 3a. The substitution r = a tan θ will prove useful in carrying out the
integration.
F=
412
11.10 EXERCISES
11.24
11.25
Prove equation (11.22) and, by taking b = zx2 i + zy 2 j + (x2 − y 2 )k, show that the
two integrals
I=
x2 dV and J =
cos2 θ sin3 θ cos2 φ dθ dφ,
both taken over the unit sphere, must have the same value. Evaluate both directly
to show that the common value is 4π/15.
In a uniform conducting medium with unit relative permittivity, charge density ρ,
current density J, electric field E and magnetic field B, Maxwell’s electromagnetic
equations take the form (with µ0 0 = c−2 )
(i) ∇ · B = 0,
(iii) ∇ × E + Ḃ = 0,
(ii) ∇ · E = ρ/0 ,
(iv) ∇ × B − (Ė/c2 ) = µ0 J.
2
The density of stored energy in the medium is given by 12 (0 E 2 + µ−1
0 B ). Show
that the rate of change of the total stored energy in a volume V is equal to
0
1
−
J · E dV −
(E × B) · dS,
µ0 S
V
11.26
where S is the surface bounding V .
[ The first integral gives the ohmic heating loss, whilst the second gives the
electromagnetic energy flux out of the bounding surface. The vector µ−1
0 (E × B)
is known as the Poynting vector. ]
A vector field F is defined in cylindrical polar coordinates ρ, θ, z by
y cos λz
F0 ρ
x cos λz
F = F0
i+
j + (sin λz)k ≡
(cos λz)eρ + F0 (sin λz)k,
a
a
a
where i, j and k are the unit vectors along the Cartesian axes and eρ is the unit
vector (x/ρ)i + (y/ρ)j.
(a) Calculate, as a surface integral, the flux of F through the closed surface
bounded by the cylinders ρ = a and ρ = 2a and the planes z = ±aπ/2.
(b) Evaluate the same integral using the divergence theorem.
11.27
The vector field F is given by
F = (3x2 yz + y 3 z + xe−x )i + (3xy 2 z + x3 z + yex )j + (x3 y + y 3 x + xy 2 z 2 )k.
11.28
Calculate (a) directly, and (b) by using Stokes’ theorem the value of the line
integral L F · dr, where L is the (three-dimensional) closed contour OABCDEO
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).
A vector force field F is defined in Cartesian coordinates by
3
2
z xy/a2
y xy/a2
x + y xy/a2
y
xy
F = F0
e
e
e
j
+
+
+
1
i
+
+
k
.
3a3
a
a3
a
a
Use Stokes’ theorem to calculate
0
F · dr,
L
where L is the perimeter of the rectangle ABCD given by A = (0, 1, 0), B = (1, 1, 0),
C = (1, 3, 0) and D = (0, 3, 0).
413