# Need help understanding why a negative work done is caused by an external force in electrostatics

I was working on a problem that included a electric dipole. I found the change in Potential energy that it took to move a proton from A to B in my diagram, however this was a negative value and I've been told that this work is due to an external force. Can someone please explain to my why its due to the "External Force"?

• Something has to "push" the proton, it's can't "roll uphill" on its own? Movement towards the + charge is "uphill" because the proton has a + charge. Is that the confusion? – JMLCarter Jan 23 '17 at 11:23
• That helped a bit but I'm wondering why negative work which is what I calculated tells us that the force is due to an external force. – Teyash Arjun Jan 23 '17 at 12:07
• The sign you get out depends on the signs of the inputs. Which is to say in some calculation there could be a positive value of work and it could still mean an external force was required. Examine your definitions and see if you can determine that energy must be input to the proton. Are you integrating from a force equation from original to final position - what is the force, that resisting or that causing motion? – JMLCarter Jan 23 '17 at 12:15
• I was just using the face that the negative change in potential energy is equal to the work done. The resisting force would be that created by the positive charge. – Teyash Arjun Jan 23 '17 at 12:23
• Whether it is negative or not depends on whether you move from high to low potential or in the other direction. – JMLCarter Jan 23 '17 at 12:52

To simplify matters suppose that there is a system which consists of two positive charges, one of which cannot move and that system of charges has some electric potential energy.
There are forces of repulsion between the two charges and these are internal forces – they are a Newton’s third law pair. You now apply a force on the mobile charge which is equal in magnitude but opposite in direction to the repulsive force on it, ie the net force on the mobile charge is zero.
The force that you apply on the mobile charge is an external force, it originates from outside the system of two charges. Now allow the mobile charge to move away from the other charge from an initial position of rest $A$ to a final position $B$ of rest.

You find that the electric potential energy of the system of two charges has decreased.
How can that happen?

It happens because the internal repulsive force does work and the work done is positive because the internal force and the displacement of the internal force are in the same direction.
As a result of that work done by the repulsive force, the electric potential energy of the system decreases.

The work done on the system by the external force exerted by you when the charge moved from position $A$ to position $B$ has the same value as the work done by the internal force except that it is negative.
It is negative because the external force and the displacement of the external force are in opposite directions.

So how does one interpret negative work done on the system by the external force exerted by you?

It is the positive work done by the system on you.

When you do positive work on a system, energy is transferred from you to the system and when you do negative work on a system, energy is transferred from the system to you.