[LECS/EM-02002] Solved Examples --- Computation of Electric Field

We will show that the electric field  due to a uniformly charged disk at a point on the axis of the disk is given by

\[\begin{eqnarray} E_z &=&\frac{2Qz}{a^2 4\pi\epsilon_0}\Big[1-\frac{z}{(z^2+a^2)^{1/2}}\Big]\\ \label{eq11A} \end{eqnarray}\] At a point on the axis of the disk, only the \(z\) component is non-zero, the other two components vanish, \(E_x=E_y\).

We will show that for a uniformly charged sphere, radius \(a\) and charge \(Q\),

the electric field is given by \[\begin{equation}  \vec E =  \begin{cases} 0 & \text{if } r<a\\      \frac{Q}{4\pi\epsilon_0 R^2 }  & \text{if } r > a  \end{cases} \end{equation}\]

The electric field due to a charged ring, at a point, \(P\) on its axis,  is computed using Coulomb's law. We will show that the electric field of uniformly charged ring, radius \(R\), at a  point on the axis of the ring,  is given by

\[\begin{equation}  \vec E = \frac{qz}{4\pi\epsilon_0(z^2+r_0^2)^{3/2}} \, \hat k. \end{equation}\] where \(q\) is the total charge and \(z\) is the distance if field point from the center of the ring.