Answer to Question #224433 in Electricity and Magnetism for Farhat Tasnim

Question

dielectric sphere of radius R is hollowed out in the region 0 ≤ r ≤ s and a

thin, grounded, conducting shell is inserted at r = s. The sphere is placed in a uniform,

external E⃗ -field E⃗ = E0ẑalong the z axis. The dielectric constant of the hollowed sphere is εr

. Calculate the potential in the region r ≥ R.

Answer 1

Answer:-

We have given data :-

region as $0\le r\le S$

r=S

and $\overrightarrow{\in}=\in_o \overrightarrow{z}$

$\in_r$ = Dielectric Constant

Assume , before hollowed out a Sphere (R1) the charge on whole sphere is 'Q'

And after hollowed out , and charged to q [ Dielectric filled in the cavity ]

$\bull$ Relation between q and Q

$\boxed{q=Q(1-\frac{1}{\in_r})}$ $\rightarrow$ this is charge left on hollow Spherical Shell.

$\bull$ Electrical Field due to'Q' at a distance 'r'

$|\in (r)|=k.\frac{Q}{r^2}$ $[\because k=\frac{1}{4\pi\in_o}]$

$\in_o =k\frac{Q}{r^2}\\ \boxed{Q=\frac{1}{k}\in_o.r^2}$

therefore , $q=\frac{\in_o}{k}r(1-\frac{1}{\in_r})$

Electric potential (V) due to charge (q) at a distance 'r' given by

$V=k.\frac{q}{r}\\ \ \ \ \ = k.\frac{\in_or}{k}(1-\frac{1}{e_r})\\$

$\therefore \boxed{V=\in_or(1-\frac{1}{e_r})}$