Figure $1$: Problem $4.$

a$.$ Consider a pair of charges $+q$ and $-q$ at a distance $d$ rotating in the $x-y$ plane at an angular frequency $.$ At time $t=0,$ the particles are oriented along the x $-$axis. Determine the time$-$dependent dipole moment $p(t),$ the associated radiation pattern $d\frac{P}{d},$ and the total power radiated $P.($Hint: express the dipole moment in terms of its Cartesian components.$)$

b$.$ In the Bohr hydrogen atom, the electron moves in a circular orbit of radius $5{10}^{-9}cm.$ Determine the radiation pattern and total power radiated by the electron as it orbits the nucleus as a function of the distance $r$ between them. You may assume the electron velocity is non$-$relativistic. $($Hint: treat the Coulomb force between the electron and nucleus as a centripetal force $F_{\text{c}\text{e}\text{n}\text{t}}=m\frac{v^2}{r}$ with centripetal acceleration $a=\frac{v^2}{r}.)$