Question

A solid sphere of radius R has a uniform charge density and total charge Q. Find the total energy of the sphere U in terms of

Answer 1

The total energy U of a uniformly charged sphere can be calculated by integrating the energy density, which relates to the square of the electric field, over the sphere's volume. Gauss's law is used to find the electric field.

Step-by-step explanation:

The student is asking about calculating the total energy U of a solid sphere with uniform charge density and a total charge Q. We begin by considering the expression for the potential energy between two point charges in spherical coordinates. We then use concepts similar to those analyzing a parallel-plate capacitor, including energy density in a region with an electrical field. For an isolated conducting sphere, we can analyze the electric field using Gauss's law, which is helpful for both uniform and non-uniform charge densities.

To find the total energy U of the sphere, one approach involves integrating the energy density over the entire volume of the sphere. This calculation requires considering the electric field inside and outside the sphere, which can be derived using Gauss's law, and then integrating the energy density, which is proportional to the square of the electric field.

It's important to consider that the expression for U may involve integral calculus and the application of Gauss's law to determine the electric field within different regions of the sphere. The principle of superposition may also be essential when examining electric fields from various charge distributions.

Answer 2

`\mathbf{U = (3 k_c Q^2_(total))/(5R)}`

Step-by-step explanation:

From the information given;

The surface area of a sphere =
`4 \pi r ^2`

If the sphere is from the collection of spherical shells of infinitesimal thickness = dr

Then,

the volume of the thickness and the sphere is;

V =
`4 \pi r ^2 \ dr`

Using Gauss Law

`V(r) = (k_cq(r))/(r)`

here,

q(r) =charge built up contained in radius r

since we are talking about collections of spherical shells, to work required for the next spherical shell r +dr is

`-dW= dU = V(r) dq = (k_c \ q(r))/(r) \ dq`

where;

`q (r) = (4)/(3) \pi r^3 \rho`

dq which is the charge contained in the next shell of charge

here dq = volume of the shell multiply by the density

`dq = 4 \pi r^2 \ dr \ \rho`

equating it all together

`dU = (k_c (4)/(3) \pi r^3 \rho)/(r) 4 \pi r^2 \ dr \ \rho = (16 \pi^2 \ k_c \ \rho^2)/(3) \ r^4 \ dr`

Integration the work required from the initial radius r to the final radius R, we get;

`U = \int^R_0 \ dU`

`U = \int^R_0 (16 \pi^2 \ k_c \rho^2)/(3) r^4 \ dr`

`U = \int^R_0 (16 \pi^2 \ k_c \rho^2)/(3) [(r^5)/(5)]^R_0`

`U = (16 \pi^2 k_c \rho^2)/(15) \ R^5`

Recall that:

the total charge on a sphere, i.e
`Q_(total) = (4)/(3) \pi R^3 \rho`

Then :

`\mathbf{U = (3 k_c Q^2_(total))/(5R)}`