Download Elementary Particles in the theory of relativity

Constant Electromagnetic Field
Section 19
Constant fields
• E and H are independent of time t.
• f and A can be chosen time independent, too.
We can add an arbitrary constant to f without changing E or H.
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Only a constant (no t or r dependence) can be added to f for constant fields.
An extra condition is usually imposed, e.g. f = 0 at infinity.
Then f is determined uniquely.
We can add an arbitrary constant to A without changing E or H,
but we can also add functions.
• A function of coordinates grad(f) can still be added to A without changing
E or H.
• A is not unique even for constant fields.
Energy of charge in constant electromagnetic field.
If fields are constant, the Lagrangian is independent of time
The energy is conserved and equals the Hamiltonian.
The constant field adds energy ef to the particle.
• ef is the “potential” energy of a charge in the field.
• The energy does not depend on A, so H does no work on the charge.
• Only E changes the energy of a particle.
Uniform constant fields
• Electric field has no r dependence
• f = -E.r
Uniform constant fields
• A is not unique
• Two examples that both give uniform H:
A1 = (1/2) H x r
A2 = [-H y, 0, 0]
• These two choices differ by grad(f), where f = -xyH/2
One possible vector potential for a uniform field
Let’s see if it works
The other possible choice of vector potential for uniform constant field was
(We chose the z-axis parallel to the H field.)
Now let’s check this one
We said they differed by
Let’s check the difference
Were two of the
possible choices for the
vector potential of a
uniform constant field
Which function f(x) vs. x does not give a
constant uniform electric field?
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2
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Which function f(x) vs. x does not give a
constant uniform electric field?
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