Question

a) Find the total electrostatic energy stored in a uniformly charged sphere (not a shell) of radius R and charge q. Express your answer in terms of q, R, and constants of nature. There are many different ways to do this, we want you to use both of the following two different methods to check yourself. (i) Figure out E(r) and then use Griffiths Eq. 2.45 shown below (Be careful to integrate over all space, not just where the charge is!). W L E?d?r Jallspace (ii) Figure out V(r), use Griffiths Eq. 2.43 shown below (What region do you need to in- tegrate this over?) w=/ pvár

Answer 1

We have to find the total electrostatic energy Csphere By using two methods (1) figuring out Ecry and then using W = to (E²d3r we know that Electric field inside sphere is given by Einside = qr where rc R. 4TE RS - Electric field outside the sphere is given by E outsed = q 2 . r>R 4IT to R2 Total Electric field is E outside Etotal = = Einside + ar + q 4TEOP2 \ 4TEOR3 aboue egn puttuy 4 T r² dr this we in get d3r by and replacing a (since dir=dv=srar - 4128 dr W = to Ciara w Seth Coast mitor) arredor hen arrange flere stvar + Sortope) 4 vºdy 4 tr ² dr JATTEOR?

(5) * (*) = 392 20 TT to R Now we will solue it by using the ii) method that is figure out very and then usind .. W = I savd²r we know here f = volume chare density 1 s= q where u= 4 Tr3 (volume of the sphere ) 3q Thus 8 = . q 4 Tr3 4 1 r3 33 potential due to a charged sphere is given by va OR? ( putting the value in the about we get W = 1 / S 3q 1 a 4 Tradr 2R 2 4 Tr3 I 4 TEOR 12 (replacing d²r by Atrider 392 = sql 16 TT to R6 (3R²_r2 r2 dr 392 r [ 3R 16T to ROLI

= 392 20 T to R. As we can see we get the same result by using different method which is assumed. Hence the total energy stored in the uniformly Charged sphere is 3q² Ans, 2 отво R