Two concentric spherical conducting shells are separated by...

Two concentric spherical conducting shells are separated by vacuum. The inner shell has total charge +Q and radius a, and outer shell has charge -Q and radius b. Using the integration of electric field energy density find the electric energy stored in the system. Take Q = 20 ?C, a = 50 cm and b = 80 cm. The energy, U0 = 1.35 Units J Using the obtained energy and formula for the energy stored in a capacitor, U = Q^2 / 2C, find the capacitance of the system. The capacitance, C0 = 14.81 Units pF Repeat the calculations for the same system with a dielectric material of ? = 1.5 inserted in between the shells. The energy, U1 = 0.9 Units J The capacitance, C1 = 22.2 Units pF

Transcript

00:01 Now in this question, basically look at a capacitor.

00:03 This capacitor consists of two concentric spherical conducting shares as shown this diagram, right? you have an auto share, you also have an inner share.

00:11 They have a radius b and a respectively.

00:14 So on the auto share, you have a negative charge minus q.

00:17 On the inner share, you have a charge q.

00:19 So the electric field obviously by symmetry exists only between the shares, like this here.

00:26 And you can also see that the electric field is symmetric.

00:30 With respect to rotations about the center of these spheres, right? so if, and you can actually work out the electric field, for example, at a point, at, well, the electric field on the sphere, on a virtual sphere, concentric with the spheres and has a radius, let's say, all right? and we can easily work out using gas law, right? because e.

00:57 Error times the flux, the electroflux, which is err times the surface area of the virtual of the virtual grain sphere with radius r is given by 4 pi r squared times er, right? and this must be equal to the total charge inside this virtual sphere and that is actually given by a q, right? so from this you find that er simply is given by q, oh sorry, it's four pi q, i forgot, four pi q, right? so you find er is given by q over r squared, right? so the energy, the total energy, you stored in this electric field, is given by 1 over 8 pi and integral over the volume and the electric field squared, right? r0, right? so, and this is given by, of course, you know, the integral is, because the electric field exists only within, only between the shares.

01:53 So the integral is only down over the volume, between the shares.

01:57 And you can, because the electric does not depend on the, does not depend on the angles.

02:03 So you can integrate over the angle, which gives you basically a four pi, right? so you get a four pi over eight pi.

02:09 And times an integral, rather integration is given by d, let's say, r prime, r prime squared from a to b...

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