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From what I understand from my notes (correct me if I'm wrong), electric potential energy is the work done in bringing the charge from infinity to a certain point from the source charge. Let this point be X. In the case of a source charge +Q, an electrostatic force of repulsion will be exerted on the test charge +q and the test charge will accelerate towards infinity and away from the source charge in an isolated system.

Near infinity or as the distance from the test charge and source charge approaches infinity, electric force and electric field strength approaches zero. To bring the test charge to point X, you would need a force exerted on the test charge that acts in the direction towards the source charge. This force is the external force which acts in the opposite direction to the electric force exerted on the test charge.

As the test charge is brought closer to the source charge, the electric force increases, which means the external force has to be made to be constantly equal to electric force in order for the test charge to continue moving towards the source charge at constant velocity. I'm assuming that it is important for the test charge to move at constant velocity, otherwise the work done on the test charge/energy input into the system will consist of both electric potential energy and kinetic energy.

The graph of Force-distance(from test charge to source charge) should look like this: enter image description here

So, work done or electric potential energy would be:

enter image description here

Here's kind of where I'm unsure about things:

1) Near infinity, the test charge would still be accelerating away from the source charge, although the electric force would be very small or close to 0. In order for the test charge to be brought in the opposite direction towards the source charge, the external force has to be more than the electric force at the point near infinity, such that the resultant force exerted on the test charge will be towards the source charge. The test charge will accelerate towards source charge until external force is equal to the electric force, in which it will then travel at constant velocity towards P if external force continues to be equal to electric force. Otherwise, if external force is constantly equal to electric force as stated in my notes, then the test charge would still continue to move away from the source according to Newton's first law. Is this correct?

2) In regards to the test charge moving at constant velocity as it is being displaced from infinity to point X, what happens if external force exerted on the test charge near infinity is equal to the electric force exerted on the test charge at point X or a Newtons (displayed in diagram)? Although the object would accelerate initially, as the magnitude of the acceleration would decrease, the KE would decrease because it would be converted into EPE. Is this another way to calculate EPE?

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Otherwise, if external force is constantly equal to electric force as stated in my notes, then the test charge would still continue to move away from the source according to Newton's first law. Is this correct?

When the net force on the test charge is zero then it is either at rest or moving with constant velocity.

You are correct that if before the external force is applied the test charge is released it will move away from source charge.
However what is wrong with the test charge having an initial velocity towards the source charge and keeping the net force on the test charge zero?

Or making the external force slight larger than the force of repulsion for a short distance when the test charge is far away from the source charge and then keeping the net force on the test charge zero.
So the "extra" amount of work done on the test charge is $\Delta F_{\rm external,far}\,\Delta r_{\rm far}$.
Then when the test charge is close to the source charge making the external force smaller than the force of repulsion so that the test charge arrives at point $X$ with zero velocity.
Overall the "reduced amount of work done on the test charge is $\Delta F_{\rm external,near}\,\Delta r_{\rm near}$ such that $\Delta F_{\rm external,far}\,\Delta r_{\rm far} + \Delta F_{\rm external,near}\,\Delta r_{\rm near}=0$ ie the net amount of work done by changing the net force on the test charge is zero.

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  • $\begingroup$ Thanks! But what about my second point, is it not feasible to let the external force be equal to the electric force at point X near infinity? $\endgroup$ – xander Jun 5 at 7:30
  • $\begingroup$ @xander Of course it is but your purpose is to find the work done by an external force in bringing the test charge nearer to the source charge without changing the kinetic energy of the test charge. So you could allow the test charge to move away from the source charge but then you would have to apply an external force greater than the force of repulsion to get the test charge moving in the correct direction ie nearer the source change. $\endgroup$ – Farcher Jun 5 at 7:38
  • $\begingroup$ My mistake; realised that even though the magnitude of acceleration would decrease as the test charge is brought closer to the source charge, the test charge would still be increasing in velocity until point X which would mean an increase in KE. You would then have to decrease the external energy to stop the test charge at P and decrease the KE back to 0. Thanks for the help! $\endgroup$ – xander Jun 5 at 7:49

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