From Static to Time-Dependent Potentials and Fields

The Helmholtz potentials (Equation (1)) were initially introduced as solutions for static fields only. How might we build on the Helmholtz expressions to develop time-dependent potentials and fields?

QUESTIONS
(Score is number right minus number wrong.)

Expressions for scalar and vector fields (E, B) taken from electrostatics and magnetostatics for distributed sources can be applied to time-dependent problems by the simple expedient of expressing the sources (charge and current) as functions of a 'retarded time' (See Equation (6)).

: True; False

Expressions for scalar and vector potentials (V, A) taken from electrostatics and magnetostatics for distributed sources can be applied to time-dependent problems by the simple expedient of expressing the sources (charge density and current) as functions of a 'retarded time' (See Equation (7)).

: True; False

Scalar and vector potentials that satisfy Maxwell's Equations in general for time-dependent, distributed sources (charge density and current) are given by

: Equation (1); Equation (7); Equation (9); Equation (11)

Electric and magnetic fields that satisfy Maxwell's Equations in general for time-dependent, distributed sources (charge density and current) are given by

: Equation (6); Equation (8); Both; Neither

A scalar potential V that satisfies Maxwell's Equations for a single, charged, moving particle can be obtained by the simple expedient of expressing the single particle static potential in terms of a 'retarded time' (Equation (9)).

: True; False

A scalar potential V that satisfies Maxwell's Equations for a single, charged, moving particle can be obtained by the simple expedient of taking the limit in Equation (7) according to the prescription in

: Equation (10a); Equation (10b); Equation (10c); None of these.

Scalar and vector potentials (V, A) that satisfy Maxwell's Equations for a single, charged, moving particle are given by

: Equation (7); Equation (9); Equation (11)

General expressions for the fields (E, B) that satisfy Maxwell's Equations for a single, charged, moving particle are given by

: Equation (6); Equation (8); Equation (12); Equation (13)

In deriving correct expressions for potentials and fields, either for distributed charge distributions or for a single, moving charged particle, one formally encounters expressions for the gradient and partial derivative with respect to time of the 'retarded time'.

Since time does not have spacial dependence, the spatial gradient of retarded time vanishes identically.

: True; False

The derivations of expressions for fields originating from a time-dependent charge distribution and from a single, moving charged particle are fundamentally different. In one case the source point does not move with time and in the second it does move with time. (See diagram).

: True; False

The correct expressions for gradient and time derivative of the retarded time for a distributed charge distribution are,

: Zero and Equation (5a); Equation (4a) and Equation (5a); Equation (4a) and Equation (5b); Equation (4b) and Equation (5a); Equation (4b) and Equation (5b)

The correct expressions for gradient and time derivative of the retarded time for a single, moving charged particle are,

: Zero and Equation (5a); Equation (4a) and Equation (5a); Equation (4a) and Equation (5b); Equation (4b) and Equation (5a); Equation (4b) and Equation (5b)

The expressions on the right-hand-side of Equation (12) (u,a, v, etc.) are to be evaluated at the present time rather than the retarded time.

: True; False

The fields in Equation (13) are a special case of Equation (12) for which we have

: constant, nonzero acceleration.; circular motion at constant speed.; constant speed, straight line.; Bremsstrahlung.

The expressions on the right-hand-side of Equation (13) (v, R, theta, etc.) are to be evaluated at the present time rather than the retarded time.

: True; False

EQUATIONS

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