# Work of external forces and electrostatic potential energy

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1. Oct 23, 2016

### crick

1. The problem statement, all variables and given/known data
Two metal spheres of equal radius $R$ are placed at big distance one from the other. Sphere 1 has total charge $q$ and sphere 2 has no charge. The two speheres are moved one towards the other until they touch, then they are moved again far away one from the other. What is the external work $W_{EXTERNAL \,\, FORCES}$ done during the process?

2. Relevant equations
Electrostatic potential energy of a sphere is
$U_{EL}=\frac{1}{2} \frac{q^2}{4 \pi \epsilon_0 R}$

3. The attempt at a solution
I was convinced that, since the only forces are the electric forces and the external forces, $$W_{ELECTRIC \,\, FORCES}= - \Delta U_{EL}= U_{EL \,\, initial}-U_{EL \,\, final}=\frac{1}{2} \frac{q^2}{4 \pi \epsilon_0 R}-2 \cdot \frac{1}{2} \frac{(q/2)^2}{4 \pi \epsilon_0 R}$$
So the work of external forces, which is what is asked, is
$$W_{EXTERNAL\,\, FORCES}= \Delta U_{EL}=U_{EL \,\, final}- U_{EL \,\, initial}=2 \cdot \frac{1}{2} \frac{(q/2)^2}{4 \pi \epsilon_0 R}-\frac{1}{2} \frac{q^2}{4 \pi \epsilon_0 R}$$

But solutions says exaclty the opposite, that is

$$W_{EXTERNAL \,\, FORCES}= - \Delta U_{EL}= U_{EL \,\, initial}-U_{EL \,\, final}=\frac{1}{2} \frac{q^2}{4 \pi \epsilon_0 R}-2 \cdot \frac{1}{2} \frac{(q/2)^2}{4 \pi \epsilon_0 R}$$

How can this be true? Am I missing something on the sign or is the solution wrong?

2. Oct 23, 2016

### haruspex

When the spheres make contact, work is lost, so you cannot get the work done by external forces from the net change in energy of the spheres.

3. Oct 23, 2016

### crick

Thanks for the reply! I see that there are some problems in that contact with the conservation of energy, but I'm a bit confused. Which work is lost when there is contact? The external forces or the electrostatic forces one?

So what part of potential energy should I consider?
After the contact the potential energy of the spheres does not change anymore, so I'm not even sure that any work is done..

4. Oct 23, 2016