$(20$ points$)$

As shown in the figure, a string of length $L$ has its upper end fixed, with a charged small ball of mass $m$ hanging from its lower end. The small ball is placed in a uniform electric field of strength $E,$ directed horizontally to the right. The string makes an angle of $$ with the vertical direction when the small ball is in equilibrium.

$($a$)$ What is the type of charge on the small ball? Find the charge carried by the small ball.

$($b$)$ If the string is displaced to increase the angle from $$ to $,$ and then the small ball is released, determine the value of $$ such that the small ball's velocity is exactly zero when string just reaches the vertical position.