WEEK 9: Classical Fields Coursework 2
1. (a) Define a conservative vector field in terms of line integrals around closed paths. {1}
(b) Verify that ∇ × (∇F ) = 0 for all scalar fields F by expanding the cross product in
Cartesian coordinates. {3}
(c) Given that the line integral of ∇F around any closed path is zero, explain why a
conservative field must have zero curl everywhere. {2}
2. Consider the vector field A = xyi + j.
H
(a) Calculate the line integral A.dl anticlockwise around the perimeter of a triangle in
the x–y plane with vertices at (0,0), (1, 2) and (0,2).1 {6}
(b) Calculate the curl of A. {2}
(c) State whether A is conservative or not. {1}
H
3. Ampère’s law in Electromagnetism states that C B.dl = µ0 I, where I is the current
enclosed by the curve C and µ0 is the permeability of a vacuum (a constant). The direction
of integration around the curve C is given by the right-hand rule. Consider an infinitely-
long thin wire aligned along the z-axis and carrying a current I in the +z direction. Gauss’
law can be used to show that the magnetic field is directed in anticlockwise circles around
the z-axis.
(a) Ampère’s law is usually used to calculate magnetic field, even though B is inside the
integral. Symmetry is used to extract B from inside the integral.
By integrating around a circular path in the x–y plane of radius ρ centred on the
wire, show that the magnetic field a distance ρ from the wire can be expressed in
cylindrical coordinates as
µ0 I
B= ϕ̂. {3} (1)
2πρ
(b) Although we don’t normally do the calculation this way, we can take a known result
for B and integrate round a less-symmetric path and show that the result is still µ0 I.
Convert the
H
result for B in equation (1) to Cartesian coordinates. Use the result to
calculate C B.dl around a square of side 2a in the x–y plane with its centre on the
wire.2 Hence verify that Ampère’s law is obeyed for this path also. {4}
(c) Because the line integrals in (a) and (b) enclose a singularity, we can’t use the result of
the line integrals to determine whether B is conservative or not (as with the discussion
in the notes relating to whether the velocity of a free vortex is conservative). Use
other information to determine whether B is conservative outside the wire. {3}
ρ̂ ρϕ̂ ẑ
a dx x 1
Z
= tan−1 and ∇×A= ∂ ∂ ∂
a2+x 2 a ρ ∂ρ ∂ϕ ∂z
Aρ ρAϕ Az
1
Later, you will see how this can also be done using Stokes’ theorem; you should explicitly calculate the line
integral here though.
2
Hint 1: If you use symmetry arguments, you won’t need to evaluate all four sides explicitly.
1. (a) Define a conservative vector field in terms of line integrals around closed paths. {1}
(b) Verify that ∇ × (∇F ) = 0 for all scalar fields F by expanding the cross product in
Cartesian coordinates. {3}
(c) Given that the line integral of ∇F around any closed path is zero, explain why a
conservative field must have zero curl everywhere. {2}
2. Consider the vector field A = xyi + j.
H
(a) Calculate the line integral A.dl anticlockwise around the perimeter of a triangle in
the x–y plane with vertices at (0,0), (1, 2) and (0,2).1 {6}
(b) Calculate the curl of A. {2}
(c) State whether A is conservative or not. {1}
H
3. Ampère’s law in Electromagnetism states that C B.dl = µ0 I, where I is the current
enclosed by the curve C and µ0 is the permeability of a vacuum (a constant). The direction
of integration around the curve C is given by the right-hand rule. Consider an infinitely-
long thin wire aligned along the z-axis and carrying a current I in the +z direction. Gauss’
law can be used to show that the magnetic field is directed in anticlockwise circles around
the z-axis.
(a) Ampère’s law is usually used to calculate magnetic field, even though B is inside the
integral. Symmetry is used to extract B from inside the integral.
By integrating around a circular path in the x–y plane of radius ρ centred on the
wire, show that the magnetic field a distance ρ from the wire can be expressed in
cylindrical coordinates as
µ0 I
B= ϕ̂. {3} (1)
2πρ
(b) Although we don’t normally do the calculation this way, we can take a known result
for B and integrate round a less-symmetric path and show that the result is still µ0 I.
Convert the
H
result for B in equation (1) to Cartesian coordinates. Use the result to
calculate C B.dl around a square of side 2a in the x–y plane with its centre on the
wire.2 Hence verify that Ampère’s law is obeyed for this path also. {4}
(c) Because the line integrals in (a) and (b) enclose a singularity, we can’t use the result of
the line integrals to determine whether B is conservative or not (as with the discussion
in the notes relating to whether the velocity of a free vortex is conservative). Use
other information to determine whether B is conservative outside the wire. {3}
ρ̂ ρϕ̂ ẑ
a dx x 1
Z
= tan−1 and ∇×A= ∂ ∂ ∂
a2+x 2 a ρ ∂ρ ∂ϕ ∂z
Aρ ρAϕ Az
1
Later, you will see how this can also be done using Stokes’ theorem; you should explicitly calculate the line
integral here though.
2
Hint 1: If you use symmetry arguments, you won’t need to evaluate all four sides explicitly.
