A small ball with charge $q=12\mathrm{.}4C$ and mass $m=0\mathrm{.}045kg$ is suspended from the ceiling by a string of length $L=2m$ and is initially at rest. A uniform horizontal electric field $E$ of magnitude $500\frac{V}{m}$ is applied to the ball$-$string system. The ball then begins to movo. Ignore air resistance.

a$.$ Suppose point $B$ is the highest point the ball can reach. Take $$ as the angle of the string with the vertical direction at point $B.$ Enter an expression for the change of gravitational potential enorgy $\frac{?}{{?}_{\frac{?}{?}}{U}_B}$ from point $A$ to point $B$ in terms of the symbols given. $(5)$

b$.$ Enter an expression for thmbolge of the electrical potential energy $U.$ from point $A$ to point $B$ in terms of the symbols given. $(4)$

c$.$ Setting these equal, deries an expression to determine the angle $.$ This can be simplified using the trig identities: $(4)$

$sin=2sin(\frac{}{2})cos(\frac{}{2})$

$cos=1-2si{n}^2(\frac{}{2})$

d$.$ Solve for the angle $.(2)$