21Ans is πεRV2.
The second explanation was just to prove that it connot be 2πεRV2,as it corresponds to total electrostatic energy.
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
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5 Answers
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.
219Energy density is given by 12ε0E2 E=-Vr Energy density=12ε0V2r2 energy is = ∫R2R(12ε0V2r24Πr2)∂r = ∫(R2R2Πε0V2)∂r=2Πε0V2R