# Confusion in electric potential energy

This is an extract from my book ... " We define the potential V at any point in an electric field as the potential energy per unit charge associate with a test charge q0 at that point: V=U/q0." I don't know why did the book bring the test charge q0 now in the frame. Shouldn't the potential be only by q. To be more specific in my question, V= Kq/r ... What is q, isn't it q which produces the field?? Can someone explain it with referring the analogous theme of force and electric field? To be more specific in my question, in our definition of potential, we said it is U per unit charge. How then U miraculously disappear in our net equation of V which is Kq/r and we can assume now that potential is created by a single charge. Wasn't it actually originated by definition by relating it to another charge. Thanks in advance

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mfb
Mentor
Shouldn't the potential be only by q
Right. And how do you measure this potential? -> with a test charge, which has a potential energy of U and a charge of q0.
What is q, isn't it q which produces the field?
It is.

How then U miraculously disappear in our net equation of V
It does not have to disappear, you can write U/q0 = Kq/r if you like.

Fine, when you've taken each part individually it makes sense. However, my confusion comes from the way we defined it. "We define the potential V at any point in an electric field as the potential energy per unit charge associated with a test charge q0 at that point" Why does the term "associate with q0" come in here since V=kq/r and q0 hasn't to be dealt with.

Another definition is given as " Potential at a certain point is the potential energy that would be associated with a unit charge placed at that point" How can we interpret this mathematically? In formula I mean. Generally, what is the exact definition of electric potential??

Doc Al
Mentor
Fine, when you've taken each part individually it makes sense. However, my confusion comes from the way we defined it. "We define the potential V at any point in an electric field as the potential energy per unit charge associated with a test charge q0 at that point" Why does the term "associate with q0" come in here since V=kq/r and q0 hasn't to be dealt with.
When you bring a test charge (q0) a distance r from a point charge (q), what is the potential energy?

Given that, what's the potential energy per unit charge? (Just divide by q0.)

WannabeNewton
Do you want the mathematical definition? Since the electric field ##E## is conservative, i.e. the work done in moving a particle in the field is independent of path, there exists a scalar field ##V## such that ##E = - \triangledown V##; ##V## is called the potential. Physically, if you have an electric field created by a charge ##Q## then the potential ##V## at a point in the field is related to the work needed to bring in a unit test charge from infinity and place it at that point in the field. You're just dividing out the test charge in the potential energy formula for the charge configuration which is basically the same as considering a unit test charge.

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Apart from maths, I need elaboration in the meaning of electric potential .. Thanks for your patience guys

sophiecentaur
Gold Member
2020 Award
Apart from maths, I need elaboration in the meaning of electric potential .. Thanks for your patience guys

Have you looked at what Wiki has to say about the definition of Potential? Your original quote from your book may have been taken a bit out of context and, possibly, part-way through the discussion of potential.
It relates to the work needed to bring a unit charge from infinity to the point in question. You need to check on your understanding of the definitions of each of the symbols used in your book, I think.

rcgldr
Homework Helper
It might help to use gravity near the surface of the earth as an analogy. Assuming g is constant near the surface of the earth (9.80665 m / s^2), then gravitational potential energy using the surface of the earth as a reference point = m g h. Gravitional potential is the potential energy per unit mass, so gravitational potential = g h. Since g can be assumed to be constant near the surface of the earth, gravitational potential is a function of height.

A similar analogy can be made for a charged object between two charged plates. Zero voltage would correspond to a point on the negatively charged plate, and increase linearly with distance towards the positively charged plage, assuming that the particle between the plates has positive charge.

So while gravitational potential is potential energy per unit mass, electrical potential or voltage, is potential energy per unit charge (relative to some reference point or surface).