Consider two concentric spherical shells, of radii a and b. Suppose the inner one carries a char

Consider two concentric spherical shells, of radii a and b....

Key Concepts

Potential Energy of Charge Distributions

Another method to determine the energy of a charge configuration is by using the formula U = (1/2)?q???, where the potential at the location of each charge is considered. This formulation is particularly useful for systems where the potential due to other charges can be calculated easily, allowing one to account for the interactions between disparate parts of the charge distribution.

Spherical Symmetry

Spherical symmetry greatly simplifies electrostatic problems, as it allows one to employ Gauss’s law effectively to determine the electric field in various regions. In cases involving concentric spherical configurations, symmetry reduces the complexity of the integrals involved in both the energy density method and the potential-based method, making the calculations more tractable.

Electrostatic Energy

Electrostatic energy represents the work necessary to assemble a given configuration of charges from infinity. In the context of continuous or discrete charge distributions, this energy can be calculated either by integrating the energy density of the electric field throughout space or by summing the potential energy contributions of the charge elements, reflecting the interactions among them.

Energy Density of the Electric Field

One common method for computing the electrostatic energy uses the energy density expression U = (??/2)?E² d³r. This approach involves determining the electric field in every region of space and then integrating its square over the volume occupied by the field, thereby capturing both the self-energy of individual charge distributions and their interaction energy.

Transcript

00:01 Hi there, so for this problem we need to consider two concentric spherical shells of radii a and b and we need to suppose that the inner the inner one carries a charge q and the other one a charge minus q.

00:18 So this is the situation that we have in here two spherical shells concentric one of them it has a radio of of a and the other one a radius of b.

00:38 The charge of the inner one is q and the charge of the other one is minus q.

00:48 Now we need to calculate the energy of this configuration.

00:53 Now for part a of this problem, we are told to use equation 2 .45.

01:02 So the equation 2 .45.

01:05 It states that the word done is equal to epsilon sub 0 divided by 2 the integral of the square of the electric field.

01:24 So with that, remember that in this case, the electric field is equal to 1 over 4 times pi times epsilon sub 0 times the charge q divided by the radius art square.

01:44 Where the radius is between the values a and b and 0 elsewhere.

01:55 So with that, we just simply substitute this expression into the work.

02:02 So we will obtain this work is simply epsilon sub 0 divided by 2.

02:09 The charge q divided by 4 times pi times epsilon sub 0, all of this to the square times the integral from a to b, of 1 over the radius squared to the square.

02:36 And in this case, we know that the differential is the value 4 times pi times the radius squared, integrated over the differential in the radius.

02:48 This 4 times pi comes from the solid angle that we know is the integral from integrated theta and fee and the sign of theta, of course, when we are using its vertical coordinates.

03:08 Now, with that set, in here, here, simplifying this expression, we will find that this is the charge q squared divided by a times pi times epsilon sub 0, the integral from a to b of 1 over the radius square.

03:34 So after we integrate this, we obtain the charge q squared divided by a times pi times epsilon sub 0 times 1 over a minus 1 over b so that's a solution for the first part of this problem now for part b we are asked about to use the equation 2 .47 and the results of example 2 .8 so with that and we will have that the work 1, and that we are going to call 1, is equal to 1 over 8 times pi times epsilon sub 0 times the charge q squared divided by, in this case a.

04:26 And the word done 2 is the same, but the only thing that changes is that this is the charge q squared divided by b...

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