ρ.
1. A vector field A is given in cylindrical polar coordinates ρ, φ, z by A = ρ2 ρ̂
(i) A cone, with axis aligned with the z direction, has base radius R and height h.
Taking the coordinate origin to be the centre of the base, and by expressing the
position vector r to the curved surface as a function of ρ and φ, show that the
vector area element on the curved surface is given by
hρ
ρ̂ρ + ρk̂ d ρ d φ
dS =
R
!
[8 marks]
A · dS over the entire closed surface.
(ii) Compute the integral
S
[6 marks]
(iii) State the divergence theorem. Using the expression for divergence in cylindrical
polar coordinates
1 ∂(ρ Aρ ) 1 ∂Aφ ∂Az
∇·A=
+
+
,
ρ ∂ρ
ρ ∂φ
∂z
compute the divergence of the vector field A.
[5 marks]
(iv) Compute the integral of ∇ · A over the enclosed volume. Compare your answers
for the surface and volume integrals, and comment.
[6 marks]
[Total 25 marks]
Module Code:PHY40004
3
Please go to the next page
2.
(i) (a) Show that ∇ · (∇ × B) = 0 for any vector field B.
(b) For the vector field
A = y 2 z 2 î − x 2 z 2 ĵ + z 4 k̂,
calculate ∇ × A.
[6 marks]
(ii) Consider the cube defined by the 8 vertices that are the 8 combinations of
x, y, z = 0 or 1, with
! the z axis in the vertical direction. Evaluate individually
the outwards flux, (∇ × A) · dS, over each of the four vertical faces of the cube,
S
and sum them.
[6 marks]
H
(iii) Calculate the line integral A · dr, anticlockwise when viewed from above the xy
plane, a) around the four edges of the bottom face of the cube, and b) around
the four edges of the top face of the cube in part (ii).
[7 marks]
(iv) State Stokes’ theorem. Explain the relation between the two line integrals, and
the sum from part (ii).
[6 marks]
[Total 25 marks]
Module Code:PHY40004
4
Please go to the next page
3.
(i) A sphere of radius R carries charge Q, uniformly distributed throughout the
body. Write down an expression for the charge density ρ in terms of Q and R.
By using Gauss’s law, and clearly stating the form and location of the closed
surfaces that are used, show that the electric field strength inside the sphere is
E(r) =
ρr
30
and that the electric field strength outside the sphere is
E(r) =
ρR 3
30 r 2
where r is the distance from the origin.
[8 marks]
(ii) Write down the general expression for the electric potential in terms of the integral of the electric field, where the potential is defined to be zero at r = ∞. Show
by integration that the electric potential outside the sphere is
V(r) =
ρR 3
.
3 0 r
[5 marks]
(iii) Consider a dielectric sphere of radius 7.0 cm carrying a uniformly distributed
charge of 8.0 × 10−7 C.
(a) Calculate the electric potential 8.0 cm from the surface of the sphere.
(b) Derive an expression for the electric potential inside the sphere. Thus calculate the potential difference between the centre and the surface of the
sphere.
[12 marks]
[Total 25 marks]
Module Code:PHY40004
5
Please go to the next page
4. Consider an electrically conducting, zero resistance rod of mass m that rests on two
long, conducting, frictionless rails as shown in the diagram below. These form a
circuit with a resistor R, switch S and a battery providing a voltage V. The volume is
filled with a uniform vertical magnetic field B; the rod has an initial speed u = 0.
(i) At time t = 0, switch S is closed. Show that the initial acceleration of the rod is
a(t = 0) =
VLB
.
Rm
In which direction does the rod accelerate?
[5 marks]
(ii) Write down Faraday’s law. By considering the forces acting on the rod, show
that its acceleration at a time t is
du(t) BLV − B 2 L 2 u(t)
=
.
dt
mR
[6 marks]
(iii) Hence derive an expression for the speed of the rod as a function of time.
A rod has a length of 10cm and mass 50g. If R = 100Ω, B = 1T and a voltage
of V = 1kV is applied, what will be the speed of the rod after 1s?
[5 marks]
(iv) Write down Ampère’s law. By considering symmetry, show that the magnetic
field at the end of a long solenoid, with N turns of wire per unit length, carrying
a current I, is given by
1
B ≈ µ0 NI.
2
You are required to make a magnetic field of 10−4 T at the end of a solenoid
which has a circular cross section of diameter 1cm and length 2m. Estimate
what length of wire is required when it will carry a current of 0.2A.
[9 marks]
[Total 25 marks]
Module Code:PHY40004
6
End of examination paper