(a) From Eq. 10.72,
E ( r , t)=\frac{q}{4 \pi \epsilon_{0}} \frac{ᴫ}{( ᴫ \cdot u )^{3}}\left[\left(c^{2}-v^{2}\right) u + ᴫ \times( u \times a )\right] (10.72)
E _{1}=\frac{(q / 2)}{4 \pi \epsilon_{0}} \frac{ᴫ}{( ᴫ \cdot u )^{3}}\left[\left(c^{2}-v^{2}\right) u +( ᴫ \cdot a ) u -( ᴫ \cdot u ) a \right] . \text { Here } u =c \hat{ ᴫ }- v , ᴫ =
l \hat{ x }+d \hat{ y }, v =v \hat{ x }, a =a \hat{ x }, \text { so } ᴫ \cdot v =l v, ᴫ \cdot a =l a, ᴫ \cdot u =c ᴫ – ᴫ \cdot v =c ᴫ -l v .
We want only the x component. Noting that u_{x}=(c / ᴫ) l-v=(c l-v ᴫ) / ᴫ , we have:
E_{1_{x}}=\frac{q}{8 \pi \epsilon_{0}} \frac{ᴫ}{(c r-l v)^{3}}\left[\frac{1}{ᴫ}(c l-v ᴫ)\left(c^{2}-v^{2}+l a\right)-a(c ᴫ-l v)\right]
=\frac{q}{8 \pi \epsilon_{0}} \frac{1}{(c ᴫ-l v)^{3}}\left[(c l-v ᴫ)\left(c^{2}-v^{2}\right)+c l^{2} a-v ᴫ l a-a c ᴫ^{2}+a l v ᴫ\right] . \text { But } ᴫ^{2}=l^{2}+d^{2}.
=\frac{q}{8 \pi \epsilon_{0}} \frac{1}{(c ᴫ-l v)^{3}}\left[(c l-v ᴫ)\left(c^{2}-v^{2}\right)-a c d^{2}\right].
F _{ self }=\frac{q^{2}}{8 \pi \epsilon_{0}} \frac{1}{(c ᴫ-l v)^{3}}\left[(c l-v ᴫ)\left(c^{2}-v^{2}\right)-a c d^{2}\right] \hat{ x } . (This generalizes Eq. 11.90.)
F _{\text {self }}=\frac{q}{2}\left( E _{1}+ E _{2}\right)=\frac{q^{2}}{8 \pi \epsilon_{0} c^{2}} \frac{\left(l c^{2}-a d^{2}\right)}{\left(l^{2}+d^{2}\right)^{3 / 2}} \hat{ x } (11.90)
\text { Now } x(t)-x\left(t_{r}\right)=l=v T+\frac{1}{2} a T^{2}+\frac{1}{6} \dot{a} T^{3}+\cdots, \text { where } T=t-t_{r}, \text { and } v , a, and \dot{a} are all evaluated at the retarded time t_{r}.
(c T)^{2}=ᴫ^{2}=l^{2}+d^{2}=d^{2}+\left(v T+\frac{1}{2} a T^{2}+\frac{1}{6} \dot{a} T^{3}\right)^{2}=d^{2}+v^{2} T^{2}+v a T^{3}+\frac{1}{3} v \dot{a} T^{4}+\frac{1}{4} a^{2} T^{4};
c^{2} T^{2}\left(1-v^{2} / c^{2}\right)=c^{2} T^{2} / \gamma^{2}=d^{2}+v a T^{3}+\left(\frac{1}{3} v \dot{a}+\frac{1}{4} a^{2}\right) T^{4} . Solve for T as a power series in d:
T=\frac{\gamma d}{c}\left(1+A d+B d^{2}+\cdots\right) \Rightarrow \frac{c^{2}}{\gamma^{2}} \frac{\gamma^{2} d^{2}}{c^{2}}\left(1+2 A d+2 B d^{2}+A^{2} d^{2}\right)=d^{2}+v a \frac{\gamma^{3} d^{3}}{c^{3}}(1+3 A d)+\left(\frac{v \dot{a}}{3}+\frac{a^{2}}{4}\right) \frac{\gamma^{4}}{c^{4}} d^{4}.
Comparing like powers of A=\frac{1}{2} v a \frac{\gamma^{3}}{c^{3}} ; 2 B+A^{2}=\frac{3 v a \gamma^{3}}{c^{3}} A+\left(\frac{v \dot{a}}{3}+\frac{a^{2}}{4}\right) \frac{\gamma^{4}}{c^{4}}.
2 B=\frac{3 v a \gamma^{3}}{c^{3}} \frac{1}{2} v a \frac{\gamma^{3}}{c^{3}}-\frac{1}{4} v^{2} a^{2} \frac{\gamma^{6}}{c^{6}}+\frac{v \dot{a}}{3} \frac{\gamma^{4}}{c^{4}}+\frac{a^{2} \gamma^{4}}{4 c^{4}}=\frac{v \dot{a}}{3} \frac{\gamma^{4}}{c^{4}}+\frac{\gamma^{6} a^{2}}{4 c^{4}}\left(\frac{1}{\gamma^{2}}-\frac{v^{2}}{c^{2}}\right)+\frac{3}{2} \frac{v^{2} a^{2} \gamma^{6}}{c^{6}}
=\frac{\gamma^{4}}{c^{4}}\left[\frac{v \dot{a}}{3}+\frac{a^{2} \gamma^{2}}{4}\left(1-\frac{v^{2}}{c^{2}}-\frac{v^{2}}{c^{2}}+6 \frac{v^{2}}{c^{2}}\right)\right] \Rightarrow B=\frac{\gamma^{4}}{2 c^{4}}\left[\frac{v \dot{a}}{3}+\frac{\gamma^{2} a^{2}}{4}\left(1+4 \frac{v^{2}}{c^{2}}\right)\right].
T=\frac{\gamma d}{c}\left\{1+\frac{v a}{2} \frac{\gamma^{3}}{c^{3}} d+\frac{\gamma^{4}}{2 c^{4}}\left[\frac{v \dot{a}}{3}+\frac{\gamma^{2} a^{2}}{4}\left(1+4 \frac{v^{2}}{c^{2}}\right)\right] d^{2}\right\}+() d^{4}+\cdots (generalizing Eq. 11.93).
T=\frac{1}{c} d+\frac{a^{2}}{8 c^{5}} d^{3}+() d^{4}+\cdots (11.93)
l=v T+\frac{1}{2} a T^{2}+\frac{1}{6} \dot{a} T^{3}+\cdots
=\frac{v \gamma d}{c}\left\{1+\frac{v a}{2} \frac{\gamma^{3}}{c^{3}} d+\frac{\gamma^{4}}{2 c^{4}}\left[\frac{v \dot{a}}{3}+\frac{\gamma^{2} a^{2}}{4}\left(1+4 \frac{v^{2}}{c^{2}}\right)\right] d^{2}\right\}+\frac{1}{2} a \frac{\gamma^{2} d^{2}}{c^{2}}\left[1+v a \frac{\gamma^{3}}{c^{3}} d\right]+\frac{1}{6} \dot{a} \frac{\gamma^{3}}{c^{3}} d^{3}
=\left(\frac{v \gamma}{c}\right) d+\frac{a}{2} \frac{\gamma^{4}}{c^{2}}\left(1-\frac{v^{2}}{c^{2}}+\frac{v^{2}}{c^{2}}\right) d^{2}+\left\{\frac{v \gamma}{2 c} \frac{\gamma^{4}}{c^{4}}\left[\frac{v \dot{a}}{3}+\frac{\gamma^{2} a^{2}}{4}\left(1+4 \frac{v^{2}}{c^{2}}\right)\right]+\frac{1}{2} a \frac{\gamma^{2}}{c^{2}} v a \frac{\gamma^{3}}{c^{3}}+\frac{1}{6} \dot{a} \frac{\gamma^{3}}{c^{3}}\right\} d^{3}
=\left(\frac{v \gamma}{c}\right) d+\left(\frac{a \gamma^{4}}{2 c^{2}}\right) d^{2}+\frac{\gamma^{3}}{2 c^{3}}\left[\frac{\dot{a}}{3}\left(1+\gamma^{2} \frac{v^{2}}{c^{2}}\right)+\frac{v \gamma^{4} a^{2}}{c^{2}}\left(\frac{1}{4}+\frac{v^{2}}{c^{2}}+1-\frac{v^{2}}{c^{2}}\right)\right] d^{3}
=\left(\frac{v \gamma}{c}\right) d+\left(\frac{a \gamma^{4}}{2 c^{2}}\right) d^{2}+\frac{\gamma^{5}}{2 c^{3}}\left[\frac{\dot{a}}{3}+\frac{5}{4} \frac{v \gamma^{2} a^{2}}{c^{2}}\right] d^{3}+() d^{4}+\cdots
ᴫ=c T=\gamma d\left\{1+\frac{v a}{2} \frac{\gamma^{3}}{c^{3}} d+\frac{\gamma^{4}}{2 c^{4}}\left[\frac{v \dot{a}}{3}+\gamma^{2} a^{2}\left(\frac{1}{4}+\frac{v^{2}}{c^{2}}\right)\right] d^{2}\right\}+() d^{4}+\cdots
c ᴫ-l v=c \gamma d+\frac{v a \gamma^{4}}{2 c^{2}} d^{2}+\frac{\gamma^{5}}{2 c^{3}}\left[\frac{v \dot{a}}{3}+\gamma^{2} a^{2}\left(\frac{1}{4}+\frac{v^{2}}{c^{2}}\right)\right] d^{3}-\frac{v^{2} \gamma}{c} d-\frac{a v \gamma^{4}}{2 c^{2}} d^{2}-\frac{\gamma^{5} v}{2 c^{3}}\left[\frac{\dot{a}}{3}+\frac{5}{4} \frac{v \gamma^{2} a^{2}}{c^{2}}\right] d^{3}+\cdots
=c \gamma d\left(1-\frac{v^{2}}{c^{2}}\right)+\frac{\gamma^{5}}{2 c^{3}}\left[\frac{v \dot{a}}{3}+\gamma^{2} a^{2}\left(\frac{1}{4}+\frac{v^{2}}{c^{2}}\right)-\frac{v \dot{a}}{3}-\frac{5}{4} \frac{v^{2} \gamma^{2} a^{2}}{c^{2}}\right] d^{3}+\cdots
=\frac{c}{\gamma} d+\frac{\gamma^{5} a^{2}}{8 c^{3}} d^{3}+() d^{4}+\cdots
c l-v ᴫ=v \gamma d+\frac{a \gamma^{4}}{2 c} d^{2}+\frac{\gamma^{5}}{2 c^{2}}\left(\frac{\dot{a}}{3}+\frac{5}{4} \frac{v \gamma^{2} a^{2}}{c^{2}}\right) d^{3}-v \gamma d-\frac{v^{2} a}{2} \frac{\gamma^{4}}{c^{3}} d^{2}-\frac{v \gamma^{5}}{2 c^{4}}\left[\frac{v \dot{a}}{3}+\gamma^{2} a^{2}\left(\frac{1}{4}+\frac{v^{2}}{c^{2}}\right)\right] d^{3}
=\frac{a \gamma^{4}}{2 c}\left(1-\frac{v^{2}}{c^{2}}\right) d^{2}+\frac{\gamma^{5}}{2 c^{2}}\left[\frac{\dot{a}}{3}+\frac{5}{4} \frac{v \gamma^{2} a^{2}}{c^{2}}-\frac{v^{2}}{c^{2}} \frac{\dot{a}}{3}-\frac{v \gamma^{2} a^{2}}{c^{2}}\left(\frac{1}{4}+\frac{v^{2}}{c^{2}}\right)\right] d^{3}+() d^{4}+\cdots
=\left(\frac{a \gamma^{2}}{2 c}\right) d^{2}+\frac{\gamma^{5}}{2 c^{2}}\left[\frac{\dot{a}}{3 \gamma^{2}}+\frac{v \gamma^{2} a^{2}}{c^{2}}\left(\frac{5}{4}-\frac{1}{4}-\frac{v^{2}}{c^{2}}\right)\right] d^{3}+() d^{4}+\cdots
=\left(\frac{a \gamma^{2}}{2 c}\right) d^{2}+\frac{\gamma^{3}}{2 c^{2}}\left(\frac{\dot{a}}{3}+\frac{v \gamma^{2} a^{2}}{c^{2}}\right) d^{3}+() d^{4}+\cdots
(c ᴫ-l v)^{-3}=\left[\frac{c d}{\gamma}\left(1+\frac{\gamma^{6} a^{2}}{8 c^{4}} d^{2}\right)\right]^{-3}=\left(\frac{\gamma}{c d}\right)^{3}\left(1-3 \frac{\gamma^{6} a^{2}}{8 c^{4}} d^{2}\right)+\cdots
F _{ self }=\frac{q^{2}}{8 \pi \epsilon_{0}}\left(\frac{\gamma}{c d}\right)^{3}\left(1-3 \frac{\gamma^{6} a^{2}}{8 c^{4}} d^{2}\right)\left\{\left[\left(\frac{a \gamma^{2}}{2 c}\right) d^{2}+\frac{\gamma^{3}}{2 c^{2}}\left(\frac{\dot{a}}{3}+\frac{v \gamma^{2} a^{2}}{c^{2}}\right) d^{3}\right] \frac{c^{2}}{\gamma^{2}}-a c d^{2}\right\} \hat{ x }
=\frac{q^{2}}{8 \pi \epsilon_{0}} \frac{\gamma^{3}}{c^{3} d}\left(1-\frac{3}{8} \frac{\gamma^{6} a^{2}}{c^{4}} d^{2}\right)\left[-\frac{a c}{2}+\frac{\gamma}{2}\left(\frac{\dot{a}}{3}+\frac{v \gamma^{2} a^{2}}{c^{2}}\right) d\right] \hat{ x }
=\frac{q^{2}}{8 \pi \epsilon_{0}} \frac{\gamma^{3}}{c^{3} d} \frac{1}{2}\left[-a c+\gamma\left(\frac{\dot{a}}{3}+\frac{v \gamma^{2} a^{2}}{c^{2}}\right) d+() d^{2}+\cdots\right] \hat{ x }
=\frac{q^{2}}{4 \pi \epsilon_{0}}\left[-\gamma^{3} \frac{a}{4 c^{2} d}+\frac{\gamma^{4}}{4 c^{3}}\left(\frac{\dot{a}}{3}+\frac{v \gamma^{2} a^{2}}{c^{2}}\right)+() d+\cdots\right] \hat{ x } (generalizing Eq. 11.95).
F _{\text {self }}=\frac{q^{2}}{4 \pi \epsilon_{0}}\left[-\frac{a}{4 c^{2} d}+\frac{\dot{a}}{12 c^{3}}+() d+\cdots\right] \hat{ x } (11.95)
Switching to \text { t: } v\left(t_{r}\right)=v(t)+\dot{v}(t)\left(t_{r}-t\right)+\cdots=v(t)-a(t) T=v(t)-a \gamma d / c . (When multiplied by d, it doesn’t matter—to this order—whether we evaluate at t or at t_{r} .)
1-\left[\frac{v\left(t_{r}\right)}{c}\right]^{2}=1-\frac{\left[v(t)^{2}-2 v a \gamma d / c\right]}{c^{2}}=\left[1-\frac{v(t)^{2}}{c^{2}}\right]\left(1+\frac{2 a v \gamma^{3} d}{c^{3}}\right) , so
\gamma=\left[1-\left(\frac{v\left(t_{r}\right)}{c}\right)^{2}\right]^{-1 / 2}=\gamma(t)\left(1-\frac{v a \gamma^{3}}{c^{3}} d\right) ; a\left(t_{r}\right)=a(t)-T \dot{a}=a(t)-\frac{\dot{a} \gamma}{c} d.
Evaluating everything now at time t:
F _{ self }=\frac{q^{2}}{4 \pi \epsilon_{0}}\left[-\gamma^{3} \frac{\left(1-3 v a \gamma^{3} d / c^{3}\right)(a-\dot{a} \gamma d / c)}{4 c^{2} d}+\frac{\gamma^{4}}{4 c^{3}}\left(\frac{\dot{a}}{3}+\frac{v \gamma^{2} a^{2}}{c^{2}}\right)+() d^{2}+\cdots\right] \hat{ x }
=\frac{q^{2}}{4 \pi \epsilon_{0}}\left[-\frac{\gamma^{3} a}{4 c^{2} d}+\frac{\gamma^{3}}{4 c^{2}}\left(\frac{\dot{a} \gamma}{c}+3 \frac{v a^{2} \gamma^{2}}{c^{3}}\right)+\frac{\gamma^{4}}{4 c^{3}}\left(\frac{\dot{a}}{3}+\frac{v \gamma^{2} a^{2}}{c^{2}}\right)+() d+\cdots\right] \hat{ x }
=\frac{q^{2}}{4 \pi \epsilon_{0}}\left[-\frac{\gamma^{3} a}{4 c^{2} d}+\frac{\gamma^{4}}{4 c^{3}}\left(\dot{a}+\frac{\dot{a}}{3}+3 \frac{v a^{2} \gamma^{2}}{c^{2}}+\frac{v \gamma^{2} a^{2}}{c^{2}}\right)+() d+\cdots\right] \hat{ x }
=\frac{q^{2}}{4 \pi \epsilon_{0}}\left[-\frac{\gamma^{3} a}{4 c^{2} d}+\frac{\gamma^{4}}{3 c^{3}}\left(\dot{a}+3 \frac{v a^{2} \gamma^{2}}{c^{2}}\right)+() d+\cdots\right] \hat{ x } (generalizing Eq. 11.96).
F _{\text {self }}=\frac{q^{2}}{4 \pi \epsilon_{0}}\left[-\frac{a(t)}{4 c^{2} d}+\frac{\dot{a}(t)}{3 c^{3}}+() d+\cdots\right] \hat{ x } (11.96)
The first term is the electromagnetic mass; the radiation reaction itself is the second term: F_{ rad }^{ int }=\frac{\mu_{0} q^{2}}{12 \pi c} \gamma^{4}\left(\dot{a}+3 \frac{v a^{2} \gamma^{2}}{c^{2}}\right) (generalizing Eq. 11.99), so the generalization of Eq. 11.100 is
F_{ rad }^{ int }=\frac{\mu_{0} q^{2} \dot{a}}{12 \pi c} (11.99)
F_{ rad }=\frac{\mu_{0} q^{2} \dot{a}}{6 \pi c} (11.100)
F_{ rad }=\frac{\mu_{0} q^{2}}{6 \pi c} \gamma^{4}\left(\dot{a}+3 \frac{v a^{2} \gamma^{2}}{c^{2}}\right).
\text { (b) } F_{\text {rad }}=A \gamma^{4}\left(\dot{a}+\frac{3 \gamma^{2} a^{2} v}{c^{2}}\right), \text { where } A \equiv \frac{\mu_{0} q^{2}}{6 \pi c} . \quad P=A a^{2} \gamma^{6} (Eq. 11.75). What we must show is that
\int_{t_{1}}^{t_{2}} F_{ rad } v d t=-\int_{t_{1}}^{t_{2}} P d t, \quad \text { or } \int_{t_{1}}^{t_{2}} \gamma^{4}\left(\dot{a} v+3 \frac{v^{2} a^{2} \gamma^{2}}{c^{2}}\right) d t=-\int_{t_{1}}^{t_{2}} a^{2} \gamma^{6} d t
(except for boundary terms—see Sect. 11.2.2).
Rewrite the first term: \int_{t_{1}}^{t_{2}} \gamma^{4} \dot{a} v d t=\int_{t_{1}}^{t_{2}}\left(\gamma^{4} v\right) \frac{d a}{d t} d t=\left.\gamma^{4} v a\right|_{t_{1}} ^{t_{2}}-\int_{t_{1}}^{t_{2}} \frac{d}{d t}\left(\gamma^{4} v\right) a d t.
\text { Now } \frac{d}{d t}\left(\gamma^{4} v\right)=4 \gamma^{3} \frac{d \gamma}{d t} v+\gamma^{4} a ; \quad \frac{d \gamma}{d t}=\frac{d}{d t}\left(\frac{1}{\sqrt{1-v^{2} / c^{2}}}\right)=-\frac{1}{2} \frac{1}{\left(1-v^{2} / c^{2}\right)^{3 / 2}}\left(-\frac{2 v a}{c^{2}}\right)=\frac{v a \gamma^{3}}{c^{2}} . So
\frac{d}{d t}\left(\gamma^{4} v\right)=4 \gamma^{3} v \frac{v a \gamma^{3}}{c^{2}}+\gamma^{4} a=\gamma^{6} a\left(1-\frac{v^{2}}{c^{2}}+4 \frac{v^{2}}{c^{2}}\right)=\gamma^{6} a\left(1+3 \frac{v^{2}}{c^{2}}\right).
\int_{t_{1}}^{t_{2}} \gamma^{4} \dot{a} v d t=\left.\gamma^{4} v a\right|_{t_{1}} ^{t_{2}}-\int_{t_{1}}^{t_{2}} \gamma^{6} a^{2}\left(1+3 \frac{v^{2}}{c^{2}}\right) d t , and hence
\int_{t_{1}}^{t_{2}} \gamma^{4}\left(\dot{a} v+\frac{3 \gamma^{2} a^{2} v^{2}}{c^{2}}\right) d t=\left.\gamma^{4} v a\right|_{t_{1}} ^{t_{2}}+\int_{t_{1}}^{t_{2}}\left[-\gamma^{6} a^{2}\left(1+3 \frac{v^{2}}{c^{2}}\right)+3 \gamma^{6} \frac{a^{2} v^{2}}{c^{2}}\right] d t=\left.\gamma^{4} v a\right|_{t_{1}} ^{t_{2}}-\int_{t_{1}}^{t_{2}} \gamma^{6} a^{2} d t . qed