Energy stored in electric field

[Fluorescent]

Homework Statement

It's told me to treat two charged concentric sphere's as parallel plates with Q=5nC on one, and -Q on the other (both with uniform spread). The distance between them is 5mm, and the surface area of each plate/sphere is 0.13m². Calculate the energy stored in the electric field between the plates?

Homework Equations

Obviously, parallel plates can use the equation E=Vd because there's an uniform electric field, but I don't know E or V so that equation is as good as useless. Do you think it expects me to manipulate capacitance equations to work out the answer?

Because they are infact charged sphere's, can I use equations which use the inverse square law in them? Or do i have to stick to the fact they are now parallel plates?

The Attempt at a Solution

So there's a uniform electric field, but I don't really understand what it means by energy within the electric field?

The following question is asking me to work out the total energy stored if this setup were a capacitor (which is easy enough, using Q=CV etc.), but not sure what equations to use if were looking at a normal uniform electric fields question?

 

[gneill]

The parallel plate capacitor interpretation of a spherical capacitor can be fairly accurate for certain arrangements of the geometry. In this case you're told to use the approximation, so be fearless! Calculate the capacitance as though the given area was the area of the rectangular plates, and 5mm is the separation.

With the capacitance calculated and given the charge on the cap, you can work out the voltage or energy just as you would normally.

 

[Fluorescent]

Thanks for the reply!

Are you sure there's not a second way of doing it because in the second part of the question it asks me to "compute the total energy stored in the capacitor". If I did it your way, surely I'd be doing the same thing for part i) and ii).

Although from the outset it's clear where the question is heading, it doesn't actually say capacitor at any point in part i), just "calculate the total energy stored in the electric field". It also mentions Energy Density of electric fields further up the question if that helps...

 

[gneill]

Thanks for the reply!

Are you sure there's not a second way of doing it because in the second part of the question it asks me to "compute the total energy stored in the capacitor". If I did it your way, surely I'd be doing the same thing for part i) and ii).

Although from the outset it's clear where the question is heading, it doesn't actually say capacitor at any point in part i), just "calculate the total energy stored in the electric field". It also mentions Energy Density of electric fields further up the question if that helps...

Without seeing the entire question to put things in context I can't tell what their intent was for solution methods.

EDIT: One idea occurs to me. If you were to use the given charge and plate area to determine the charge density on each plate then you could, by assuming a uniform field between them, use the formula for the field produced by a uniform sheet of charge to determine the field strength between the plates. Then the energy density will be given by the volume integral (over the volume between the plates):

$$U = \int_V \frac{1}{2}\epsilon_o |E|^2 dV$$

 

[rwisz]

I would say that the energy stored in an electric field is the potential energy of the system. In this case, the system consists of two charged concentric spheres, and the energy stored is the potential energy of the electric field between them.

To calculate this energy, we can use the equation U = (1/2)QV, where U is the potential energy, Q is the charge, and V is the potential difference between the two spheres. Since the spheres are treated as parallel plates, we can use the equation V = Ed, where E is the electric field and d is the distance between the plates.

To determine the electric field, we can use the equation E = kQ/r^2, where k is the Coulomb's constant, Q is the charge on one sphere, and r is the distance between the centers of the spheres. Since the spheres have a uniform spread of charge, we can assume that the charge is evenly distributed on the surface of each sphere, and thus the charge on each plate is Q/2.

Plugging in the values given in the problem, we get:
E = (9x10^9 Nm^2/C^2)(5x10^-9 C)/(.005m)^2 = 1.8x10^12 N/C
V = (1.8x10^12 N/C)(.005m) = 9x10^9 V
Q = 5x10^-9 C
d = .005m

Plugging these values into the equation for potential energy, we get:
U = (1/2)(5x10^-9 C)(9x10^9 V) = 22.5x10^-9 J = 2.25x10^-8 J

Therefore, the energy stored in the electric field between the two spheres is 2.25x10^-8 J. This is the potential energy of the system, which is the energy stored in the electric field. It is important to note that this is the potential energy of the electric field, not the total energy of the system. The total energy of the system would also include the kinetic energy of the spheres and any other forms of potential energy (such as gravitational potential energy).