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Energy stored in an electrostatic field exam question An isolated sphere has radius R and relative permittivity r. The sphere carries charge Q, which is uniformly distributed throughout the sphere’s volume. a) Show that the radial component of electric field intensity at a radius r from the sphere’s centre is: E Qr 4 r 0 R 2 (r R) b) Derive an expression for the energy stored in the electrostaic field: i) within the sphere, and ii) external to the sphere. c) Show that the total energy stored is greater than would be stored by a sphere of the same radius carrying the same charge uniformly distributed over its surface by a factor of 1 5 r Solution a) For the uniformly-charged sphere, the charge contained within a spherical volume of radius r is: Qr Qr 3 R3 (r R) the electric flux density Dr is uniform over the surface area (4r2) of the spherical volume and from Gauss’s Law: Qr Qr 3 Qr 3 Qr 2 4 r D D r r 3 2 3 R 4r R 4R 3 E D Er r 0 Qr 4r 0 R 3 b) Energy stored per unit volume = ½DE i) Now consider a spherical shell of radius r and thickness r. Volume = 4r2.r and the energy stored in the shell is: energy = ½DE. 4r2.r 1 Qr Qr Q2r 4 2 . . 4 r r .r 2 4R 3 4 R 3 8r 0 R 6 r 0 so the total energy stored within the sphere of radius R is: R R Q2r 4 Q2 1 r6 internal energy . dr 6 6 5 0 8 R 8 R 0 ii) r 0 r 0 Q2 40r 0 R External to the sphere: Q 4r 2 Q Er 40 r 2 Q 4r 2 Dr Dr E D 0 Now for a spherical shell of radius r (r>R) and thickness r. Volume = 4r2.r and the energy stored in the shell is: energy = ½DE. 4r2.r 1 Q Q Q2 2 . . 4 r r .r 2 4r 2 4 r 2 80 r 2 0 so the total energy stored outside the sphere of radius R is: Q2 Q2 Q2 1 . dr external energy 2 r R 80 80 R R 8 0 r c) Total energy stored by the uniformly-charged sphere = internal energy + external energy = Q2 40r 0 R Q2 80 R Q2 40 r 0 R 1 5 r For the surface-charged sphere, the internal field is zero, so there is zero internal energy stored. The analysis for the external energy stored is the same as above, so energy stored = Q2 80 R Q2 1 5 r 40 r 0 R 1 1 Ratio of energy stored = 2 5 r Q 80 R