*q*

_{1}can be said to exert a force

**F**on a separate test charge

*q*

_{2}.

Similarly, a direct approach to electric potential energy is that a source charge

*q*

_{1}can be said to store electric potential energy

*EPE*on with a separate test charge charge

*q*

_{2}.

*EPE*

*changes*in this electric potential energy are caused by moving the source charge and test charge closer together, or moving them farther apart.

*EPE*can be increased (thus making ∆

*EPE*positive) by pushing together like-sign charges (positive-positive, or negative-negative), or pulling apart opposite-sign charges (positive-negative, or negative-postive). This is because it takes work to push together like-sign charges (which repel, and don't want to be pushed together); or takes work to pull apart opposite-sign charges (which attract, and don't want to be pulled apart).

On the other hand, work is done by charges that are allowed to do what they want to, thus decreasing

*EPE*(and making ∆

*EPE*negative), by letting like-sign charges (positive-positive, or negative-negative) move farther apart from each other (because these charges repel); or letting opposite-sign charges (positive-negative, or negative-positive) move closer to each other (because these charges attract).

*q*

_{1}and test charge

*q*

_{2}at a certain distance

*r*from each other can be calculated by an equation that looks similar to the magnitude of the electric force (but note the

*r*

^{–1}dependence for electric potential energy, as opposed to the

*r*

^{-2}dependence of electric forces). With the constant

*k*, units of coulombs

^{2}and meters cancel, leaving units of N·m, which is what we have seen last semester as joules.

(Note that this equation calculates the value of electric potential energy

*EPE*of two charges at a fixed separation distance from each other. In order to find the ∆

*EPE*change in electric potential energy, you would need to calculate the initial

*EPE*and final

_{i}*EPE*electric potential energies of the two charges at their initial

_{f}*r*and final

_{i}*r*separation distances.)

_{f}*Q*can be said to create an electric field

**E**everywhere around itself, and then this electric field exerts a force

**F**on a separate test charge

*q*.

*Q*can be said to create a potential

*V*everywhere around itself, and this potential stores electric potential energy

*EPE*with a separate test charge charge

*q*.

*Q*creates a potential

*V*everywhere around it. The magnitude of this potential can be calculated for a location at a distance

*r*from the source charge, and has units of joules per coulomb (J/C), which is often re-expressed as volts (V). The sign of the potential depends on the sign of the source charge

*Q*: if the source charge is positive, then the values of the potential everywhere around it are positive; if the source charge is negative, then the values of the potential everywhere around it are negative.

*r*from the source charge must have the same value of potential, such that these circles are often referred to as

*equipotentials*.

Note that the direction of electric field lines is indicative of the relative values of the equipotentials: for the positive source charge, each subsequent outer equipotential corresponds to smaller and smaller positive values of potential (due to the

*r*

^{–1}dependence); for the negative source charge, each subsequent inner potential equipotential corresponds to larger and larger negative values of potential. So in either case, electric field lines point towards

*decreasing*electric potential values.

*q*. In this equation, the electrical potential energy

*EPE*stored by the test charge

*q*is the test charge

*q*multiplied by the potential

*V*at the test charge's location. Since the test charge

*q*could be positive or negative, and the potential

*V*could have positive or negative values, then the electrical potential energy stored would have the correct sign depending the the product of these two signs.

(Notice that coulombs (C) multiplied by volts (V, or J/C) results in units of joules (J). In this sense potential or "voltage" can be said to be "potential" potential energy, that is, a location in space will only have a value for potential (measured in volts, or joules per coulomb), but any test charge

*q*placed at this location in space will then have a value for electric potential energy (measured in joules) given by the product of the potential and amount of charge placed there.)

*which would also point in the direction of decreasing potential and decreasing potential energy*; while putting a negative test charge in the presence of these electrical fields and potentials will cause an electrical force to be exerted in the opposite direction of a field line,

*which would point in the direction of increasing potential, but decreasing potential energy*.

In any case, as a check the direction of the force on any source charge

*q*should be attractive or repulsive depending on whether it has the opposite or same sign as the source charge

*Q*, and also check that the source charge

*q*moves in the direction that would decrease its electric potential energy. However, positive test charges move in the direction of decreasing potential, while negative test charges move in the direction of increasing potential! (If all this sounds confusing, at least be assured that these rules are consistent.)

*Q*as a "peak." Here we have a positive test charge +

*q*as a hapless snowmobiler on this "peak." Initially as the snowmobiler rides up the "peak," increasing his potential and increasing his potential energy, work is required to push it "uphill" (provided by an external source, here the snowmobile engine). Left alone, the snowmobile and snowmobiler will slide "downhill," in the direction of decreasing potential and decreasing potential energy (as we have seen before, for two positive charges separating from each other). What would happen if the snowmobiler were a

*negative*test charge -

*q*? Then we would observe the very odd behavior of this negative snowmobiler sliding "uphill," in the direction in

*increasing*potential--but this would be the direction of

*decreasing*potential energy, as we have seen before, for a negative charge and a positive charge getting closer to each other.

What would happen if there was a negative source charge -

*Q*creating a "well?" Then a positive test charge +

*q*snowmobiler, left alone, would slide "downhill," in the direction of decreasing potential and decreasing potential energy (as we have seen before, for a negative charge and positive charge getting closer to each other). Then consider the very odd case of a

*negative*test charge -

*q*snowmobiler sliding on this "well"--we would observe the very odd behavior of this negative snowmobiler sliding "uphill," in the direction in

*increasing*potential--but this would be the direction of

*decreasing*potential energy, as we have seen before, for two negative charges separating from each other. (Video source: "Grazy hill climb crash with go pro cam.")

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