91Ó°ÊÓ

An insulating circular ring (radius b) lies in the xy plane, centered at the origin. It carries a linear charge density $\mathit{λ}\mathbf{=}{\mathit{λ}}_{\mathbf{0}}\mathit{s}\mathit{i}\mathit{n}\mathit{ϕ}$, where${\mathit{λ}}_{\mathbf{0}}$ is constant and$\mathit{ϕ}$ is the usual azimuthal angle. The ring is now set spinning at a constant angular velocity Ӭ about the z axis. Calculate the power radiated

A current $\mathbf{I}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$flows around the circular ring in Fig. 11.8. Derive the general formula for the power radiated (analogous to Eq. 11.60), expressing your answer in terms of the magnetic dipole moment, $\mathbf{m}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ , of the loop.

Find the angle ${\mathit{θ}}_{\max }$ at which the maximum radiation is emitted, in Ex. 11.3 (Fig. 11.13). Show that for ultra relativistic speeds ( $\mathit{Ï\dots }$close to$\mathrm{c}$), ${\mathit{θ}}_{\max }\mathbf{â‰\dots }\sqrt{\mathbf{(}\mathbf{1}\mathbf{−}\mathbf{β}\mathbf{)}\mathbf{/}\mathbf{2}}$. What is the intensity of the radiation in this maximal direction (in the ultra relativistic case), in proportion to the same quantity for a particle instantaneously at rest? Give your answer in terms of$\mathit{γ}$.

(a) A particle of charge $q$moves in a circle of radius$R$at a constant speed$v$. To sustain the motion, you must, of course, provide a centripetal force$\frac{m{v}^2}{R}$what additional force ($F_e$) must you exert, in order to counteract the radiation reaction? [It's easiest to express the answer in terms of the instantaneous velocity$v$.] What power ($P_e$) does this extra force deliver? Compare$P_e$with the power radiated (use the Larmor formula).

(b) Repeat part (a) for a particle in simple harmonic motion with amplitudeand angular frequency:$\mathrm{Ó\neg }$.$\mathrm{Ó\neg }\left(t\right)=\mathbf{Acos}\mathbf{(}\mathrm{Ó\neg }t\mathbf{)}\stackrel{⌢}{z}$ Explain the discrepancy.

(c) Consider the case of a particle in free fall (constant acceleration$g$). What is the radiation reaction force? What is the power radiated? Comment on these results.

A point charge q, of mass m, is attached to a spring of constant $\mathit{k}\mathbf{.}{\mathit{Y}}^{\mathbf{2}}\mathbf{<}\mathbf{<}{\mathit{Ó\neg }}_{\mathbf{0}}\mathbf{}\mathit{A}\mathit{t}\mathbf{}\mathit{t}\mathit{i}\mathit{m}\mathit{e}\mathbf{}\mathit{t}\mathbf{=}\mathbf{0}$it is given a kick, so its initial energy is ${\mathit{U}}_{\mathbf{0}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{m}{\mathit{v}}_{\mathbf{0}}^{\mathbf{2}}$. Now it oscillates, gradually radiating away this energy.

(a) Confirm that the total energy radiated is equal to U₀. Assume the radiation damping is small, so you can write the equation of motion as and the solution as

role="math" localid="1658840767865" $\mathit{x}\mathbf{+}\mathit{y}\mathbf{+}\mathit{x}\mathbf{+}{\mathit{Ó\neg }}_{\mathbf{0}}^{\mathbf{2}}\mathit{x}\mathbf{=}\mathbf{0}\mathbf{,}$

and the solution as

$\mathit{x}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{v}}_{\mathbf{0}}}{{\mathbf{Ó\neg }}_{\mathbf{0}}}{\mathit{e}}^{\mathbf{-}\mathbf{y}\mathbf{t}\mathbf{/}\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}\left({\mathrm{Ó\neg }}_{0}t\right)$

with ${\mathit{Ó\neg }}_{\mathbf{0}}\mathbf{≡}\sqrt{\mathbf{k}\mathbf{/}\mathbf{m}}\mathbf{,}\mathit{Y}\mathbf{=}{\mathit{Ó\neg }}_{\mathbf{0}}^{\mathbf{2}}\mathit{T}$, and ${\mathit{Y}}^{\mathbf{2}}\mathbf{<}\mathbf{<}{\mathit{Ó\neg }}_{\mathbf{0}}$ (drop Y²in comparison to ${\mathit{Ó\neg }}_{\mathbf{0}}^{\mathbf{2}}$, and when you average over a complete cycle, ignore the change in ${\mathit{e}}^{\mathbf{-}\mathbf{y}\mathbf{Ï„}}$).

(b) Suppose now we have two such oscillators, and we start them off with identical kicks. Regardless of their relative positions and orientations, the total energy radiated must be 2U₀. But what if they are right on top of each other, so it's equivalent to a single oscillator with twice the charge; the Larmor formula says that the power radiated is four times as great, suggesting that the total will be 4U₀. Find the error in this reasoning, and show that the total is actually2U₀, as it should be.

An electric dipole rotates at constant angular velocity $\mathbf{Ó\neg }$in the$\mathbf{xy}$ plane. (The charges,$\mathbf{Â\pm }\mathbf{q}$ , are at ${\mathbf{r}}_{\mathbf{Â\pm }}\mathbf{=}\mathbf{Â\pm }\mathbf{R}\mathbf{(}\mathbf{cosÓ\neg t}\widehat{\mathbf{x}}\mathbf{+}\mathbf{sinÓ\neg t}\widehat{\mathbf{y}}\mathbf{)}$; the magnitude of the dipole moment is $\mathbf{p}\mathbf{=}\mathbf{2}\mathbf{qR}$.)

(a) Find the interaction term in the self-torque (analogous to Eq. 11.99). Assume the motion is nonrelativistic ( $\mathbf{Ó\neg R}\mathbf{<}\mathbf{<}\mathbf{c}$).

(b) Use the method of Prob. 11.20(a) to obtain the total radiation reaction torque on this system. [answer: $\mathbf{-}\frac{{\mathbf{μ}}_{\mathbf{0}}{\mathbf{p}}^{\mathbf{2}}{\mathbf{Ó\neg }}^{\mathbf{3}}}{\mathbf{6}\mathbf{Ï€³¦}}\widehat{\mathbf{z}}$]

(c) Check that this result is consistent with the power radiated (Eq. 11.60).

As a model for electric quadrupole radiation, consider two oppositely oriented oscillating electric dipoles, separated by a distance $d$, as shown in figure in Fig. 11.19. Use the results of Sect. 11.1.2 for the potentials of each dipole, but note that they are not located at the origin. Keeping only the terms of first order in $d$:

(a) Find the scalars and vector potentials

(b) Find the electric and magnetic fields.

(c) Find the pointing vector and the power radiated

An ideal electric dipole is situated at the origin; its dipole moment points in the z direction and is quadratic in time:

$\mathbf{p}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\stackrel{\mathbf{¨}}{\mathbf{p}}}_{\mathbf{0}}{\mathbf{t}}^{\mathbf{2}}\widehat{\mathbf{z}}\mathbf{\text{ â¶Ä‰â¶Ä‰â¶Ä‰}}\mathbf{(}\mathbf{−}\mathbf{∞}\mathbf{<}\mathbf{t}\mathbf{<}\mathbf{∞}\mathbf{)}$

where${\stackrel{\mathbf{¨}}{\mathbf{p}}}_{\mathbf{0}}$is a constant.

1. Use the method of Section 11.1.2 to determine the (exact) electric and magnetic fields for all r > 0 (there's also a delta-function term at the origin, but we're not concerned with that).
2. Calculate the power, $\mathit{P}\mathbf{(}\mathit{r}\mathbf{,}\mathit{t}\mathbf{)}$, passing through a sphere of radius r.
3. Find the total power radiated (Eq. 11.2), and check that your answer is consistent with Eq. 11.60.21

we calculated the energy per unit time radiated by a (non-relativistic) point charge- the Larmor formula. In the same spirit:

(a) Calculate the momentum per unit time radiated.

(b) Calculate the angular momentum per unit time radiated.

8 Suppose the (electrically neutral) yz plane carries a time-dependent but uniform surface current K (t) Z.

(a) Find the electric and magnetic fields at a height x above the plane if

(i) a constant current is turned on at t = 0:

$\mathit{K}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{\left\{}\begin{array}{l}\mathbf{0}\mathbf{,}\mathbf{\text{ â¶Ä‰â¶Ä‰â¶Ä‰â€‰}}\mathbf{t}\mathbf{≤}\mathbf{0}\\ {\mathbf{K}}_{\mathbf{0}}\mathbf{,}\mathbf{\text{ â¶Ä‰â¶Ä‰}}\mathbf{t}\mathbf{>}\mathbf{0}\end{array}\mathbf{\right\}}$

(ii) a linearly increasing current is turned on at t = 0:

$\mathit{K}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{\left\{}\begin{array}{l}\mathbf{0}\mathbf{,}\mathbf{\text{ â¶Ä‰â¶Ä‰â¶Ä‰â€‰}}\mathbf{t}\mathbf{≤}\mathbf{0}\\ \mathbf{Î\pm }\mathbf{t}\mathbf{,}\mathbf{\text{ â¶Ä‰â¶Ä‰}}\mathbf{t}\mathbf{>}\mathbf{0}\end{array}\mathbf{\right\}}$

(b) Show that the retarded vector potential can be written in the form, and from

$\mathit{A}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{=}\frac{{\mathbf{μ}}_{\mathbf{0}}\mathbf{c}}{\mathbf{2}}\widehat{\mathbf{z}}\underset{0}{\overset{\mathrm{∞}}{∫}}K(t−\frac{x}{c}−u)\mathit{d}\mathit{u}$

And from this determine E and B.

(c) Show that the total power radiated per unit area of surface is

$\frac{{\mathbf{μ}}_{\mathbf{0}}\mathbf{c}}{\mathbf{2}}{\mathbf{\left[}\mathbf{K}\mathbf{\right(}\mathbf{t}\mathbf{\left)}\mathbf{\right]}}^{\mathbf{2}}$

Explain what you mean by "radiation," in this case, given that the source is not localized.22