Question

If the electric field of a point charge were proportional to 1>r3 instead of 1>r2, would Gauss’s law still be valid? Explain your reasoning. (Hint: Consider a spherical Gaussian surface cantered on a single point charge.)

Short answer

No, The Gauss’s law won’t be valid if the electric field is proportional to$\frac{1}{{\mathrm{r}}^3}$

Step-by-step solution

Relation between electric flux and electric field

The electric field equation when$\mathrm{E\alpha }=\frac{1}{{\mathrm{r}}^3}$will be$\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{\mathrm{q}}{{\mathrm{r}}^3}$

Now from gauss law we know that the relation between electric flux and the electric field and the charge is: role="math" localid="1663933522142" $$\begin{gathered} {\infty }_{\mathrm{E}}=\mathrm{EA} \\ =\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{\mathrm{q}}{{\mathrm{\epsilon }}_0}\left(4{\mathrm{\pi r}}^2\right) \\ =\frac{\mathrm{q}}{{\mathrm{r\epsilon }}_0} \end{gathered}$$

Therefore, it is proved that gauss law is not valid when electric field is proportional to$\frac{1}{{\mathrm{r}}^3}$