Vector Potential: Retarded Time
An infinite straight wire carries the uniform (inside the wire) current shown in Equation (4) in such a way that the charge density of the wire is zero. What is the electric field E(s,t) at point P?
QUESTIONS
(Score is number right minus number wrong.)
Since the net charge density vanishes everywhere along the wire, the electric potential V has no sources, vanishes everywhere, and the electric field must also vanish everywhere. There is only a magnetic field at point P.
True
False
There will be an electric field at point P if there is a vector potential A(s,t) that varies with time.
True
False
Immediately when the current begins to flow (t=0), a vector potential A is created at all points in space.
True
False
'Information' about changes in sources (currents and charge densities) travels to distant points in empty space at the speed of light, c.
True
False
The vector potential A(s,t) at point P is zero until at least the time given by
Equation (5a)
Equation (5b)
Equation (5c)
Information from the element of wire dz' cannot arrive at point P until at least the time given by
Equation (5a)
Equation (5b)
Equation (5c)
The length of the wire that contributes to the vector potential at point P for some finite time (t > s/c) is determined by the limits of integration. We could use the limits of integration indicated in
Equation (6a)
Equation (6b)
Equation (6c)
Equation (6d)
It is plausible that the Helmholtz result (Equation (2b)) for the vector potential, derived for magnetostatics, can be made time dependent by making the substitution,
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
The 'retarded time' is defined by
Equation (8a)
Equation (8b)
Equation (8c)
Equation (8d)
For a current that is confined to a line, in Equation (2b) we should substitute
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)
For the current in the integral of Equation (2b), we should substitute,
Equation (10a)
Equation (10b)
Equation (10c)
Equation (10d)
In Equation (2b) we should make the substitution,
Equation (11a)
Equation (11b)
Equation (11c)
Equation (11d)
For times t > s/c, the vector potential is given by Equation (12)
True
False
According to Equation (3), at point P for times t < s/c,
Equation (13a)
Equation (13b)
Equation (13c)
Equation (13d)
According to Equation (3), at point P for times t > s/c,
Equation (13a)
Equation (13b)
Equation (13c)
Equation (13d)
EQUATIONS
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