Download PHYS-2100 Introduction to Methods of Theoretical Physics Fall 1998 1) 2)

PHYS-2100 Introduction to Methods of Theoretical Physics
Fall 1998
Homework Assignment Due Thursday, Oct.15
3
2
1) Show explicitly that ∇ ( 1 ⁄ r ) = – r ⁄ r = – r̂ ⁄ r . Class notes from Oct.1 might be useful.
2) Calculate the static electric potential and field far from an electric dipole (Fletcher Fig.1.15):
r – ( d ⁄ 2 )k̂
z
+q
r
d
-q
r + ( d ⁄ 2 )k̂
x,y
a) What is the potential φ ( r ) in terms of the vectors shown in the figure.
b) For r » d , show that r ± ( d ⁄ 2 )k̂ ≈ r [ 1 ± ( d ⁄ r ) ( r̂ ⋅ k̂ ) ] 1 ⁄ 2 .
c) Taylor expand for small values of ( d ⁄ r ) to show that the electric potential is
1 p ⋅ r̂
1
1 p⋅r
- = ------------ --------- = – p ⋅ ∇  ---
φ ( r ) = ------------ -------- r
4πε 0 r 2
4πε 0 r 3
where p ≡ qdk̂ is called the electric dipole moment.
1 3 ( p ⋅ r̂ )r̂ – p
d) Show that E = – ∇φ ( r ) = ------------ -----------------------------in the region r » d .
3
4πε 0
r
e) Sketch the field vectors E in the xz plane (i.e. for y = 0 ).
3) In this problem you will find the properties of an electromagnetic plane wave propagating in an
arbitrary direction in free space, where that direction is given by the wave vector k ≡ kk̂ .
a) For a vector field of the form F = F 0 f ( k ⋅ r ) , where F 0 is a constant vector and f ( u )
is an arbitrary function of a single variable, show that ∇ ⋅ F = 0 implies that k ⋅ F 0 = 0 .
b) For electromagnetic waves with E = E 0 sin ( k ⋅ r – ωt ) and B = B 0 sin ( k ⋅ r – ωt ) ,
use Maxwell’s Equations to show that the waves are “transverse”. That is, show that they
propagate in a direction that is perpendicular to the direction of the fields E and B .
c) Use Maxwell’s Equations to show that B 0 = ( k̂ ⁄ c ) × E 0 . Explain why this means that
E , B , and k̂ are always mutually perpendicular.