Question

maximum value of electric intensity due to charged sphere is at

Answer 1

Electric field intensity due to charged sphere is given by
$\bold{E_{r}=\large\frac{\rho_{0}}{\in_{0}}\;[\large\frac{r}{3}-\large\frac{r^2}{4R}]}$
For finding maximum electric field intensity , differentiate it with respect to r.
$\bold{\frac{dE_r}{dr}=\frac{\rho_0}{\in_0}\:[\large\frac{1}{3}-\large\frac{2r}{4R}]}$
Now, dEr/Dr = 0
Then, 1/3 = 2r/4R
⇒R = 2R/3
At r = 2R/3 find d²Er/dr² , you will get d²Er/dr² < 0
Hence, at r = 2R/3 electric field is maximum.
so, maximum electric field intensity , Emax = $=\bold{\large\frac{\rho_{0}}{\in_{0}}\;[\large\frac{2R}{9}-\large\frac{4}{9}\;\large\frac{R^2}{4R}]}$
$\bold{=\large\frac{\rho_{0} R}{\in_{0}}\;[\large\frac{2}{9}-\large\frac{1}{9}]}$
= $\bold{= \large\frac{\rho_{0} R}{9 \in_{0}}\;}$


Hence, answer is $\bold{E_{r}|_{max} = \large\frac{\rho_{0} R}{9 \in_{0}}}$

Answer 2

Answer: infinity

Explanation:

The electric field intensity is inversely proportional to length .