10.11 EXERCISES
(a) For cylindrical polar coordinatesρ, φ, z, evaluate the derivatives of the three
unit vectors with respect to each of the coordinates, showing that only∂ˆeρ/∂φ
and∂ˆeφ/∂φare non-zero.
(i) Hence evaluate∇^2 awhenais the vectorˆeρ, i.e. a vector of unit magnitude
everywhere directed radially outwards and expressed byaρ=1,aφ=
az=0.
(ii) Note that it is trivially obvious that∇×a= 0 and hence that equation
(10.41) requires that∇(∇·a)=∇^2 a.
(iii) Evaluate∇(∇·a) and show that the latter equation holds, but that[∇(∇·a)]ρ=∇^2 aρ.(b) Rework the same problem in Cartesian coordinates (where, as it happens,
the algebra is more complicated).10.19 Maxwell’s equations for electromagnetism in free space (i.e. in the absence of
charges, currents and dielectric or magnetic media) can be written
(i) ∇·B=0, (ii)∇·E=0,
(iii) ∇×E+∂B
∂t= 0 , (iv)∇×B−1
c^2∂E
∂t= 0.
A vectorAis defined byB=∇×A,andascalarφbyE=−∇φ−∂A/∂t. Show
that if the condition(v)∇·A+1
c^2∂φ
∂t=0
is imposed (this is known as choosing the Lorentz gauge), thenAandφsatisfy
wave equations as follows:(vi)∇^2 φ−1
c^2∂^2 φ
∂t^2=0,
(vii)∇^2 A−1
c^2∂^2 A
∂t^2= 0.
The reader is invited to proceed as follows.(a) Verify that the expressions forBandEin terms ofAandφare consistent
with (i) and (iii).
(b) Substitute forEin (ii) and use the derivative with respect to time of (v) to
eliminateAfrom the resulting expression. Hence obtain (vi).
(c) Substitute forBandEin (iv) in terms ofAandφ. Then use the gradient of
(v) to simplify the resulting equation and so obtain (vii).10.20 In a description of the flow of a very viscous fluid that uses spherical polar
coordinates with axial symmetry, the components of the velocity fielduare given
in terms of thestream functionψby
ur=1
r^2 sinθ∂ψ
∂θ,uθ=− 1
rsinθ∂ψ
∂r.
Find an explicit expression for the differential operatorEdefined by
Eψ=−(rsinθ)(∇×u)φ.The stream function satisfies the equation of motionE^2 ψ= 0 and, for the flow of
a fluid past a sphere, takes the formψ(r, θ)=f(r)sin^2 θ. Show thatf(r)satisfies
the (ordinary) differential equation
r^4 f(4)− 4 r^2 f′′+8rf′− 8 f=0.