3.1 Retarded Potentials

Question 1

How fast can information travel?

Answer 1

At the speed of light

Question 2

How is the retarded time calculated?

Answer 2

By subtracting the time taken for information to travel (r-r’) / c from the observation time

Question 3

How can the retarded time change for different events observed at the same time?

Answer 3

It depends on how far those events are away from the point of observation (P)

Question 4

What is the advanced time?

Answer 4

Something happens at the observation point which influences the source point
- t + (r-r’) / c

Question 5

What is the equation of the dipole moment?

Answer 5

• p(t)* = q(t) b

- b is the separation

Question 6

What is the assumption of a short dipole?

Answer 6

The length of the dipole, b &laquo_space;λ

Question 7

What is the far field assumption?

Answer 7

Observation distance r&raquo_space; b&raquo_space; λ

Question 8

What is a Hertzian dipole?

Answer 8

The retarded time is the same across the dipole

- Also short dipole and far field

Question 9

What is the notation for a quantity to be evaluated at the retarded time?

Answer 9

It is enclosed in [ ]

Question 10

What is the time component of the 4 vector potential equal to?

Answer 10

ϕ/c

Question 11

What is the time like component equal to for a Hertzian dipole

Answer 11

µ/(4 π r ) * [p dot] cos(θ)

Question 12

What is the 4 vector potential equal to for the Hertzian dipole?

Answer 12

A^µ = (ϕ/c , z)

= { µ/(4 π r ) * [p dot] } ( cos(θ) , z)

Question 13

What would a sketch of the vector potential A(r) against r look like?

Answer 13

A sine wave with a 1/r decaying envelope

Question 14

In the Hertzian dipole in spherical coordinates, which terms are not equal to 0 for the E and B field?

Answer 14

E_θ = { µ/(4 π r ) * [p dot dot] } sin(θ)
B_ϕ = { µ/(4 π r c ) * [p dot dot] } sin(θ)

Question 15

What wave do the E_θ and B_ϕ components form and state their properties

Answer 15

They form a spherical wave proportional to sin(θ) around the z axis

• Max at pi/2 ie in the equitorial plane
• 0 at the poles

Question 16

What is the energy density equal to in electrostatics?

Answer 16

U = 1/2 epsilon_0 E^2 + 1/2 B^2 / mu_0

- If theyre equal it is just epsilon_0 c E^2 in the r direction

Question 17

What shape does the Poynting vector sketch about the z axis?

Answer 17

An infinity shape with N(theta) always pointing in the r direction

Question 18

What is the instantaenous power radiated equal to and how is it calculated?

Answer 18

Integration of N.dS over a sphere radius r

P= [p dot dot]^2 / (6 pi epsilon_0 c^3)

Question 19

How is the time average of the power radiated calculated?

Answer 19

1/2 the instantaneous power multiplied by the time averaged [p dot dot]
< P >= < [p dot dot]^2 > / (12 pi epsilon_0 c^3)