91Ó°ÊÓ

Question: A time-dependent point charge q(t) at the origin, $\mathit{ÒÏ}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}\mathbf{q}\mathbf{\left(}\mathbf{t}\mathbf{\right)}{\mathit{δ}}^{\mathbf{3}}\mathbf{\left(}\mathbf{r}\mathbf{\right)}$, is fed by a current , $\mathbf{J}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}\mathbf{-}\mathbf{\left(}\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{4}$}\right.\mathbf{Ï€}\mathbf{\right)}\mathbf{\left(}\raisebox{1ex}{$\mathbf{q}$}\!\left/ \!\raisebox{-1ex}{${\mathbf{r}}^{\mathbf{2}}$}\right.\mathbf{\right)}\hat{\mathbf{r}}$ where $\mathit{q}\mathbf{=}\raisebox{1ex}{$\mathbf{dq}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{dt}$}\right.$.

(a) Check that charge is conserved, by confirming that the continuity equation is obeyed.

(b) Find the scalar and vector potentials in the Coulomb gauge. If you get stuck, try working on (c) first.

(c) Find the fields, and check that they satisfy all of Maxwell's equations. .

Confirm that the retarded potentials satisfy the Lorenz gauge condition.

$\mathbf{∇}\mathbf{â‹\dots }\mathbf{\left(}\frac{J}{r}\mathbf{\right)}\mathbf{=}\frac{1}{r}\mathbf{(}\mathbf{∇}\mathbf{â‹\dots }\mathbf{J}\mathbf{)}\mathbf{+}\frac{1}{2}\mathbf{(}{\mathbf{∇}}^{\mathbf{\text{'}}}\mathbf{â‹\dots }\mathbf{J}\mathbf{)}\mathbf{−}{\mathbf{∇}}^{\mathbf{\text{'}}}\mathbf{â‹\dots }\mathbf{\left(}\frac{J}{r}\mathbf{\right)}$

Where $\mathbf{∇}$denotes derivatives with respect to, and${\mathbf{∇}}^{\mathbf{\text{'}}}$ denotes derivatives with respect to${\mathbf{r}}^{\mathbf{\text{'}}}$. Next, noting that $\mathbf{J}\mathbf{(}{\mathbf{r}}^{\mathbf{\text{'}}}\mathbf{,}\mathbf{t}\mathbf{−}\mathbf{r}\mathbf{/}\mathbf{c}\mathbf{)}$depends on ${\mathrm{r}}^{\text{'}}$both explicitly and through, whereas it depends on r only through, confirm that

$\mathbf{∇}\mathbf{â‹\dots }\mathbf{J}\mathbf{=}\mathbf{−}\frac{1}{c}\stackrel{\mathbf{Ë™}}{\mathbf{J}}\mathbf{â‹\dots }\mathbf{(}\mathbf{∇}\mathbf{r}\mathbf{)}$, ${\mathbf{∇}}^{\mathbf{\text{'}}}\mathbf{â‹\dots }\mathbf{J}\mathbf{=}\mathbf{−}\stackrel{\mathbf{Ë™}}{\mathbf{ÒÏ}}\mathbf{−}\frac{1}{c}\stackrel{\mathbf{Ë™}}{\mathbf{J}}\mathbf{â‹\dots }\mathbf{\left(}{\mathbf{∇}}^{\mathbf{\text{'}}}\mathbf{r}\mathbf{\right)}$

Use this to calculate the divergence of$\mathbf{A}$ (Eq. 10.26).]

(a) Suppose the wire in Ex. 10.2 carries a linearly increasing current

$\mathbf{I}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{kt}$

for$\mathbf{t}\mathbf{>}\mathbf{0}$ . Find the electric and magnetic fields generated.

(b) Do the same for the case of a sudden burst of current:

$\mathit{I}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathit{q}}_{\mathbf{0}}\mathit{δ}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$

Question: Suppose a point charge q is constrained to move along the x axis. Show that the fields at points on the axis to the right of the charge are given by

$\mathit{E}\mathbf{=}\frac{\mathbf{q}}{\mathbf{4}\mathbf{Ï€}{\mathbf{Î\mu }}_{\mathbf{0}}}\frac{\mathbf{1}}{{\mathbf{r}}^{\mathbf{2}}}\frac{\left(c+v\right)}{\left(c-v\right)}\hat{\mathbf{x}}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathit{B}\mathbf{=}\mathbf{0}$

(Do not assume is constant!) What are the fields on the axis to the left of the charge?

(a) Use Eq. 10.75 to calculate the electric field a distanced from an infinite straight wire carrying a uniform line charge .$\mathrm{λ}$, moving at a constant speed down the wire.

(b) Use Eq. 10.76 to find the magnetic field of this wire.

Question: Suppose you take a plastic ring of radius and glue charge on it, so that the line charge density is . Then you spin the loop about its axis at an angular velocity . Find the (exact) scalar and vector potentials at the center of the ring. [Answer:]

Check that the potentials of a point charge moving at constant velocity (Eqs. 10.49 and 10.50) satisfy the Lorenz gauge condition (Eq. 10.12).

For the configuration in Ex. 10.1, consider a rectangular box of length $l$, width $\mathbf{w}$, and height $\mathbf{h}$, situated a distanced $\mathbf{d}$above the $\mathbf{yz}$plane (Fig. 10.2).

Figure 10.2

(a) Find the energy in the box at time${\mathbf{t}}_1=\mathbf{d}/\mathbf{c}$, and at${\mathbf{t}}_2=(\mathbf{d}+\mathbf{h})/\mathbf{c}$.

(b) Find the Poynting vector, and determine the energy per unit time flowing into the box during the interval${\mathbf{t}}_1<\mathbf{t}<{\mathbf{t}}_2$.

(c) Integrate the result in (b) from ${\mathbf{t}}_1$to ${\mathbf{t}}_2$, and confirm that the increase in energy (part (a)) equals the net influx.

A particle of charge $q_1$is at rest at the origin. A second particle, of charge$q_2$ , moves along the axis at constant velocity $v$.

(a) Find the force $F_{12}\left(t\right)$ of${\mathit{q}}_{\mathbf{1}}$ on $q_2$, at time$t$ . (When$q_2$ is at $z=vt$).

(b) Find the force $F_{21}\left(t\right)$of$q_2$ on$q_1$ , at time $t$. Does Newton's third law hold, in this case?

(c) Calculate the linear momentum$p\left(t\right)$ in the electromagnetic fields, at time$t$ . (Don't bother with any terms that are constant in time, since you won't need them in part (d)). [Answer:$({\mathrm{μ}}_0{q}_1{q}_2/4\mathrm{π}t)$ ]

(d) Show that the sum of the forces is equal to minus the rate of change of the momentum in the fields, and interpret this result physically.

Develop the potential formulation for electrodynamics with magnetic charge (Eq. 7.44).