The potential due to a point charge q at the origin may be written as v=q4πϵ0r=q4πϵ0x2+y2+z2√ part a calculate ex using equation ex=−∂v∂x.

The electric potential due to a point charge q at the origin is given by
[tex]v= \frac{q}{4 \pi \epsilon _{0} r} = \frac{q}{4 \pi \epsilon _{0} \sqrt{x^{2}+y^{2}+z^{2}} } [/tex]

The x-component of electric field is
[tex]e_{x} = - \frac{\partial v}{\partial x} = -\frac{q}{4 \pi \epsilon {0}}(- \frac{1}{2}) \frac{2x}{(x^{2}+y^{2}+z^{2})^{3/2}} = \frac{qx}{4 \pi \epsilon _{0} (x^{2}+y^{2}+z^{2})^{3/2}} [/tex]

Answer:
[tex]e_{x} = \frac{qx}{4 \pi \epsilon _{0} (x^{2}+y^{2}+z^{2})^{3/2}} [/tex]