A metal sphere of radius R is charged to a potential V . FIND THE ELECTROSTATIC ENERGY STORED IN THE ELECTRIC FIELD WITHIN A CONCENTRIC SPHERE OF RADIUS 2R?
ANSWER GIVEN- 2Ï€Æ RV2
MY ANSWER-Ï€ÆRV2
:P
21Energy stored per unit volume=1/2*εE2
E=kQ/r2=kQ/R*(R/r2)=VR/r2
Total energy=R∫2R1/2*ε*V2R2/r4*4πr2dr=2πεV2R2R∫2R1/r2dr
=πεRV2
From another perspective:
The PE in charging the sphere to potential V is stored in the form of electric field.
V=kQ/R
PE=0∫Qkq/R*∂q=kQ2/2R=k*(VR/K)2/2R=V2R/2k=2πεV2R
Total energy stored in electric field=2πεV2R.
Hence in a concentric sphere of radius R energy must be less than the total.
21Ans is πεRV2.
The second explanation was just to prove that it connot be 2πεRV2,as it corresponds to total electrostatic energy.
9Energy density is given by 12ε0E2 E=-Vr Energy density=12ε0V2r2 energy is = ∫R2R(12ε0V2r24Πr2)∂r = ∫(R2R2Πε0V2)∂r=2Πε0V2R
