\begin{problem*}{23.59}
A charged cork ball of mass $1.00\U{g}$ is suspended on a light string
in the presence of a uniform electric field as shown in Figure P23.59.
When $\vect{E}=(3.00\ihat+5.00\jhat)\E{5}\U{N/C}$, the ball is in
equilibrium at $\theta=37.0\dg$.  Find \Part{a} the charge on the ball
and \Part{b} the tension in the string.
\end{problem*}

\begin{solution}
\Part{a}
Drawing a free-body diagram for the ball,
\begin{center}
\begin{asy}
import Mechanics;
import ElectroMag;

real theta = 37;
real E_x = 3cm / 8;
real E_y = 5cm / 8;
real L = 2cm;

real x = L*Sin(theta);
real y = -L*Cos(theta);
real E_mag = length((E_x, E_y));
real E_dir = degrees((E_x, E_y));
real T_mag = E_x/Sin(theta);
real G_mag = E_y+E_x/Tan(theta);

draw((x,y)--(0,0));
draw((0,y)--(0,0), dashed);
dot((0,0));

Angle t = Angle((0,y), (0,0), (x,y), "$\theta$");
t.draw();

Charge a = pCharge((x,y));

Vector T = Vector(a.center, mag=T_mag, dir=90+theta, "$T$");
T.draw();
Vector G = Vector(a.center, mag=G_mag, dir=-90, "$mg$");
G.draw();
Vector E = Vector(a.center, mag=E_mag, dir=E_dir, "$F_E$");
E.draw();

a.draw();

draw_ijhat((-0.7*x,y));
\end{asy}
\end{center}

Balancing forces on the ball,
\begin{align}
  0 &= \sum F_x = qE_x - T\sin(\theta) \\
  T &= \frac{qE_x}{\sin(\theta)} \\
  0 &= \sum F_y = qE_y + T\cos(\theta) - mg
    = qE_y + \frac{qE_x}{\sin(\theta)}\cos(\theta) - mg
    = qE_y + qE_x\cot(\theta) - mg \\
  q [E_y + E_x\cot(\theta)] &= mg \\
  q &= \frac{mg}{E_y + E_x\cot(\theta)}
    = \ans{1.09\E{-8}\U{C} = 10.9\U{nC}} \;.
\end{align}

\Part{b}
Plugging back in for $T$,
\begin{equation}
  T = \frac{qE_x}{\sin(\theta)} = \ans{5.44\U{mN}} \;.
\end{equation}

\end{solution}
