Work of external forces and electrostatic potential energy

[crick]

Homework Statement

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?

Homework Equations

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

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 exactly 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?

 

[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.

 

[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..

 

[haruspex]

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..

I've thought about this some more, and now think I was not helpful.
In a practical setting there would be some work lost, but in a theoretical situation, with no leakage right up until the instant of contact, there might not be. When in contact, obviously there will be Q/2 on each, in some symmetrical distribution. Perhaps the distribution immediately prior to contact is the same, with the addition of Q/2 on the charged sphere as almost a point charge at the place contact is about to be made, and a corresponding -Q/2 on the other. This would give a constant potential on each sphere, and in the limit the two would be the same.
When contact occurs, the two almost point charges neutralise, but since they had hardly any separation scarcely any work is lost.

So, back to your question. Clearly, the energy of the system has reduced, so the work done by the external forces must be negative. This agrees with your solution. But I note the wording:

What is the external work ... done during the process?

I can read that either way: the work done by external forces or the work done by the system on external entities. Is that the exact wording?