Capacitance in concentric shells

[OptimusPwn]

Homework Statement

Consider four concentric spherical metal
shells of radii r, 2r, 3r, and 4r. The middle
two shells are connected by wire, as shown.
What is the total capacitance of the system?
In your solution, clearly explain your method.

Homework Equations

u= 1/2cv^2

The Attempt at a Solution

So my teacher told us to assume we can hook it up to a battery(voltage) and that the two inner shells have the same charge and to evaluate the potential inner to outer, but I have no idea how to do this.

 

[vela]

There's a number of details you'll have to figure out for yourself, but here's generally how to approach the problem. Assume there is a charge +Q on the inner sphere. Use Gauss's law to calculate the electric field between the spheres. You'll have to figure out what the charge of the other spheres as well. Once you know the electric field, you can integrate it to calculate the potential difference between the spheres.

 

[OptimusPwn]

So when I do that I get

E(r)= Q/4(pi)(R)²Epilsonknot


E(2r)=Q/4(pi)(2R)²Epilsonknot


E(3r)=Q/4(pi)(3R)²Epilsonknot

E(4r)=Q/4(pi)(4R)²Epilsonknot


Then I use V=Ed so I integrate this?

What am I integrating though, the difference between the electric fields or the distance?
I am stuck beyond this point and are my electric fields right?

 

[vela]

I think you need to go back and review how the electric field and potential difference are related (beyond just plucking formulas out of the book) and then reread my previous post.

Your electric fields are not correct. You want the electric field for the regions between the spheres. Also, you can't use V=Ed here.

 

[rdl]

I would approach this problem using the fundamental equation for capacitance, which is C = Q/V, where C is the capacitance, Q is the charge stored, and V is the voltage applied. In this case, we are looking for the total capacitance of the system, which can be calculated by adding the individual capacitances of each shell.

First, we need to determine the charge stored on each shell. Since the two inner shells are connected by a wire, they will have the same charge. Let's call this charge Q1. The outermost shell will have a charge of Q2 since it is not connected to any other shell.

Next, we can use the equation C = Q/V to calculate the capacitance of each shell. For a spherical shell, the capacitance can be expressed as C = 4πε0r, where ε0 is the permittivity of free space and r is the radius of the shell.

Therefore, the capacitance of the innermost shell, C1, can be written as C1 = 4πε0r. Since the two inner shells are connected, their total capacitance, C2, can be calculated by adding the capacitance of each shell, which gives us C2 = 4πε0(2r) + 4πε0(3r) = 20πε0r.

Finally, the total capacitance of the system, Ctotal, can be found by adding C1 and C2, which gives us Ctotal = C1 + C2 = 4πε0r + 20πε0r = 24πε0r.

In summary, the total capacitance of the concentric shells system is 24πε0r. This method can be applied to any number of concentric shells, as long as they are connected in series.