Change in electrostatic energy on two spheres

[Marcus95]

Homework Statement

We have a spehere of radius $r_1$ and on of $r_2$ far away from each other. The first sphere has a charge $Q$. What is the change in electro static energy when they are connected together?

Homework Equations

Potential of a charged sphere: $V = \frac{Q}{4\pi\epsilon_0 r}$
Elctrostatic energy: $W = q \Delta V$

The Attempt at a Solution

Initially, the potential of the system is: $V = \frac{Q}{4\pi\epsilon_0 r_1}$
When the spheres are connected together, they become equipotential and will have charges:
$Q_1 = Q \frac{r_1}{r_1+r_2}$ and $Q_2 = Q \frac{r_2}{r_1+r_2}$. Hence the change in voltage is: $\Delta V = \frac{Q}{4\pi\epsilon_0 r_1} - \frac{Q_1}{4\pi\epsilon_0 r_1} = \frac{Q}{4\pi\epsilon_0} (\frac{r_2}{r_1(r_1+r_2)})$. So the change in potential energy is: $W=\frac{Q^2}{4\pi\epsilon_0} (\frac{r_2}{r_1(r_1+r_2)})$.

However, this is wrong. The answer is supposed to be: $W=\frac{Q^2}{8\pi\epsilon_0} (\frac{r_2}{r_1(r_1+r_2)})$. Where do I loose the factor of 1/2?

 

[TSny]

I don't follow how you can write the change in potential energy as $W$ as $Q \Delta V$. This would imply that all of the charge $Q$ undergoes the same change in potential $\Delta V$. Also, your expression for $\Delta V$ does not appear to account for the change in potential of the second sphere.

As you noted, the potential $V$ at the surface of an isolated sphere of charge $Q$ and radius $r$ is $V = \frac{Q}{4 \pi \epsilon_0 r}$ (taking $V = 0$ at inifinity). But the potential energy of the sphere is not $W = QV$.

The potential energy of an isolated sphere can be derived using integration of $dW = V \, dq$, where $q$ varies between $0$ and $Q$ and $V$ is the potential when the sphere has a charge $q$.

 

[Marcus95]

I don't follow how you can write the change in potential energy as $W$ as $Q \Delta V$. This would imply that all of the charge $Q$ undergoes the same change in potential $\Delta V$. Also, your expression for $\Delta V$ does not appear to account for the change in potential of the second sphere.

As you noted, the potential $V$ at the surface of an isolated sphere of charge $Q$ and radius $r$ is $V = \frac{Q}{4 \pi \epsilon_0 r}$ (taking $V = 0$ at inifinity). But the potential energy of the sphere is not $W = QV$.

The potential energy of an isolated sphere can be derived using integration of $dW = V \, dq$, where $q$ varies between $0$ and $Q$ and $V$ is the potential when the sphere has a charge $q$.

How do I account for the potential of the second sphere in my equation of the potential change?
Also, I see you point with the integral, thank you very much. But if I am to perform it, what potential should I calculate with? The potential of the first or second sphere?

 

[TSny]

You have found the final charge of each sphere correctly.

The spheres are so far apart that you may consider the total energy of the system be the sum of the energy of each sphere alone. So, you just need to know how to calculate the energy of an isolated sphere in terms of its charge and radius. You can do this by integration as mentioned. Or, if you are familiar with the capacitance of an isolated spherical conductor, then you can use the formula for the energy of a capacitor.

 

[Marcus95]

You have found the final charge of each sphere correctly.

The spheres are so far apart that you may consider the total energy of the system be the sum of the energy of each sphere alone. So, you just need to know how to calculate the energy of an isolated sphere in terms of its charge and radius. You can do this by integration as mentioned. Or, if you are familiar with the capacitance of an isolated spherical conductor, then you can use the formula for the energy of a capacitor.

Oh I see, so we don't have to integrate at all! Am I rigth in that the potential energy of a charged sphere is simply its potential multiplied by its charge? I tried to apply:
$\Delta V = \frac{Q^2}{4\pi\epsilon_0 r_1} - \frac{Q_1^2}{4\pi\epsilon_0 r_1}- \frac{Q_2^2}{4\pi\epsilon_0 r_2}$ but ended up with the exact same answer. :(

 

[haruspex]

the potential energy of a charged sphere is simply its potential multiplied by its charge?

No. Imagine letting a small charge dq from the sphere go to infinity. The work done is Vdq. But now the charge on the sphere is a bit less.
As all the charge leaks away, what is the average potential?

 

[Buffu]

When the spheres are connected together, they become equipotential and will have charges

Are the spheres conductors ?

I also have a similar problem, so I am asking.

 

[haruspex]

Are the spheres conductors ?

I also have a similar problem, so I am asking.

If they are not then, trivially, nothing changes, so it is safe to assume they are.

 

[Buffu]

If they are not then, trivially, nothing changes, so it is safe to assume they are.

If they are insulators then charge won't flow from shere $r_1$ to sphere $r_2$, then the surface won't become equipotential right ?

 

[haruspex]

If they are insulators then charge won't flow from shere $r_1$ to sphere $r_2$, then the surface won't become equipotential right ?

Right.

 

[Buffu]

Right.

Then why they are trivally same ?

 

[haruspex]

Then why they are trivally same ?

I didn't say they were. I wrote that when you connect them nothing changes.