$[10$ pts$]$ A parallel plate capacitor is oriented vertically as shown. A $35\mathrm{.}0$ g ball hangs from a string attached to the negatively$-$charge plate. The string makes an angle of $30$ degrees from the plate. The charge density on the plates is $30\frac{C}{m^2}.$

a$.$ Indicate on the diagram the direction of the electric field.

b$.$ Draw a fully$-$labelled free$-$body diagram for the ball in the space below. $($ Do not draw it on top of the original diagram!$).$

c$.$ What is the sign of the charge on the ball? Explain your reasoning in terms of the direction of the electric field indicated in part a$).$

The ball is negatively charged since it is being attracted by the positive plate and going against the electric field of the $2$ plates.

d$.$ Determine the magnitude of the charge on the balL