Question

An infinite number of electrons are placed along x-axis at x = 2, x = 4, x = 6, . Determine the electric field at the origin due to these charges.

Answer 1

Electric field at a point from the source charge is
$E=\frac{kq}{r}$

If electron is placed at x = 2, x=4, x=6 and so on.. .. then net electric field is given by:

$$\begin{gathered} E=\frac{kq}{2}+\frac{kq}{4}+\frac{kq}{6}------ \\ E=kq\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}----\right) \\ thisistheinfinteGPwithcommonrationR=\frac{1}{2}andfirstterm\left(a\right)=1/2 \\ Sumoftheinfinteseries=\frac{a}{1-R}whenR<1 \\ E=\frac{kqa}{1-R} \\ E=\frac{kq{\displaystyle \frac{1}{2}}}{1-{\displaystyle \frac{1}{2}}} \end{gathered}$$
E = kq, where q is the charge on the electron.
​E = 9 X 10⁹ X 1.6 X 10^-19 N/C
E = 1.44 X 10^-9 N/C