An expanding sphere, radius R(t) = υt (t > 0, constant υ) carries a charge Q, uniformly distributed over its volume. Evaluate the integral
Q_{ eff }=\int \rho\left( r , t_{r}\right) d \tauwith respect to the center. Show that Q_{ cff } \approx Q\left(1-\frac{3 v}{4 c}\right), \text { if } v \ll c.
For a point at the center, t_{r}=t-r / c , so
\rho\left( r , t_{r}\right)= \begin{cases}\frac{3 Q}{4 \pi v^{3}(t-r / c)^{3}}=\frac{3 Q}{4 \pi(v / c)^{3}(c t-r)^{3}}, & \left(r<v(t-r / c) \Rightarrow r<\frac{v t}{1+v / c}\right) \\ 0, & (\text { otherwise })\end{cases}Let a \equiv v t /(1+v / c) ; then
Q_{ eff }=\frac{3 Q}{4 \pi(v / c)^{3}} \int_{0}^{a} \frac{1}{(c t-r)^{3}} 4 \pi r^{2} d r=\frac{3 Q}{(v / c)^{3}} \int_{0}^{a} \frac{r^{2}}{(c t-r)^{3}} d r=-\left.\frac{3 Q}{(v / c)^{3}}\left[\ln (c t-r)+\frac{2 c t}{(c t-r)}-\frac{(c t)^{2}}{2(c t-r)^{2}}\right]\right|_{0} ^{a}
=-\frac{3 Q}{(v / c)^{3}}\left[\ln (c t-a)+\frac{2 c t}{(c t-a)}-\frac{(c t)^{2}}{2(c t-a)^{2}}-\ln (c t)-2+\frac{1}{2}\right]
=\frac{3 Q}{(v / c)^{3}}\left[\frac{3}{2}+\ln \left(\frac{c t}{c t-a}\right)-2\left(\frac{c t}{c t-a}\right)+\frac{1}{2}\left(\frac{c t}{c t-a}\right)^{2}\right].
\text { Now, } c t-a=c t-\frac{v t}{1+v / c}=\frac{c t}{1+v / c}\left(1+\frac{v}{c}-\frac{v}{c}\right)=\frac{c t}{1+v / c}, \text { so } \frac{c t}{c t-a}=1+\frac{v}{c} , and hence
Q_{ eff }=\frac{3 Q}{(v / c)^{3}}\left[\frac{3}{2}+\ln \left(1+\frac{v}{c}\right)-2\left(1+\frac{v}{c}\right)+\frac{1}{2}\left(1+\frac{v}{c}\right)^{2}\right]
=\frac{3 Q}{(v / c)^{3}}\left[\frac{3}{2}+\ln \left(1+\frac{v}{c}\right)-2-2 \frac{v}{c}+\frac{1}{2}+\frac{v}{c}+\frac{1}{2}\left(\frac{v}{c}\right)^{2}\right]=\frac{3 Q}{(v / c)^{3}}\left[\ln \left(1+\frac{v}{c}\right)-\frac{v}{c}+\frac{1}{2}\left(\frac{v}{c}\right)^{2}\right].
\text { If } \epsilon \ll 1, \text { then } \ln (1+\epsilon)=\epsilon-\frac{1}{2} \epsilon^{2}+\frac{1}{3} \epsilon^{3}-\frac{1}{4} \epsilon^{4}+\ldots, \text { so for } v \ll c,
Q_{ eff } \approx \frac{3 Q}{(v / c)^{3}}\left[\frac{v}{c}-\frac{1}{2}\left(\frac{v}{c}\right)^{2}+\frac{1}{3}\left(\frac{v}{c}\right)^{3}-\frac{1}{4}\left(\frac{v}{c}\right)^{4}-\frac{v}{c}+\frac{1}{2}\left(\frac{v}{c}\right)^{2}\right]=Q\left(1-\frac{3 v}{4 c}\right).