Problem 10.3 (a) Find the fields, and the charge and current distributions, corresponding to 1 qt V (r, t) = 0, A(r, 1) = 47 €0 r² Don't forget that the potentials are not necessarily given in Loren gauge and the equations you should use are 10.4 and 10.5. The potential representation (Eqs. 10.2 and 10.3) automatically fulfills the two homogeneous Maxwell equations, (ii) and (iii). How about Gauss's law (i) and the Ampère/Maxwell law (iv)? Putting Eq. 10.3 into (i), we find that v?v +(v · A) at (10.4) this replaces Poisson's equation (to which it reduces in the static case). Putting Eqs. 10.2 and 10.3 into (iv) yields a²A V x (V × A) = µoJ – Ho60V () - at or, using the vector identity V × (V × A) = V(V · A) – V²A, and rearranging the terms a bit: (va-, a²A v²A – Ho€o- av v (v.A+ Ho€o H60 ) -HoJ. (10.5) Equations 10.4 and 10.5 contain all the information in Maxwell's equations.
Problem 10.3 (a) Find the fields, and the charge and current distributions, corresponding to 1 qt V (r, t) = 0, A(r, 1) = 47 €0 r² Don't forget that the potentials are not necessarily given in Loren gauge and the equations you should use are 10.4 and 10.5. The potential representation (Eqs. 10.2 and 10.3) automatically fulfills the two homogeneous Maxwell equations, (ii) and (iii). How about Gauss's law (i) and the Ampère/Maxwell law (iv)? Putting Eq. 10.3 into (i), we find that v?v +(v · A) at (10.4) this replaces Poisson's equation (to which it reduces in the static case). Putting Eqs. 10.2 and 10.3 into (iv) yields a²A V x (V × A) = µoJ – Ho60V () - at or, using the vector identity V × (V × A) = V(V · A) – V²A, and rearranging the terms a bit: (va-, a²A v²A – Ho€o- av v (v.A+ Ho€o H60 ) -HoJ. (10.5) Equations 10.4 and 10.5 contain all the information in Maxwell's equations.
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Transcribed Image Text:I. PROBLEM 1. CHARGE AND CURRENT DISTRIBUTIONS.
Problem 10.3
(a) Find the fields, and the charge and current distributions, corresponding to
V (r, t) = 0,
A(r, t) =
4T €, r2
Don't forget that the potentials are not necessarily given in Loren gauge and the equations
you should use are 10.4 and 10.5.
The potential representation (Eqs. 10.2 and 10.3) automatically fulfills the two
homogeneous Maxwell equations, (ii) and (iii). How about Gauss's law (i) and
the Ampère/Maxwell law (iv)? Putting Eq. 10.3 into (i), we find that
a
1
v?v +(v · A) =
--p;
(10.4)
at
€0
this replaces Poisson's equation (to which it reduces in the static case). Putting
Eqs. 10.2 and 10.3 into (iv) yields
a?A
V x (V x A) = HoJ – Hoco V
at
or, using the vector identity V x (V x A) = V(V · A) – V²A, and rearranging
the terms a bit:
a²A
av
V - A + Ho€0-
at
v²A
:-HoJ.
(10.5)
Equations 10.4 and 10.5 contain all the information in Maxwell's equations.
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