This subsubtopic focuses on the equation used to calculate the electric potential energy of a two-charge system and on how charge and separation control the result.

Core relationship

Electric potential energy describes the energy associated with the electrical interaction of a pair of charged objects. For this subsubtopic, the important idea is that the energy of the two-object system is set by three things: the value of each charge and the distance between them.

For a system made of exactly two charged objects, the electric potential energy follows one general algebraic relationship.

Two point charges separated by a distance $r$ (with the separation explicitly shown) illustrate the basic geometry behind $U=\dfrac{kq_1q_2}{r}$. The diagram emphasizes that the relevant distance is the straight-line separation between the charges in the two-charge system. Source

This relationship is especially important because it shows that the sign of the charges matters just as much as their sizes.

This is a scalar equation, so the result has magnitude and sign, but no direction.

Interpreting each part of the equation

Charge magnitude

The electric potential energy depends on the product $q_1q_2$.

• If the magnitude of either charge increases, the magnitude of $U$ increases.

• If one charge is doubled while the other charge stays the same, $U$ doubles.

• If both charges are doubled, $U$ becomes four times as large because the product becomes four times larger.

This makes the equation very useful for comparison questions. You do not always need a full numerical calculation; often, a change in one variable tells you immediately how $U$ changes.

Separation

The electric potential energy is inversely proportional to the separation $r$.

Two like charges shown at two different separations (1.0 m and 0.50 m) visually demonstrate the $1/r$ dependence: decreasing $r$ increases $U$ for same-sign charges. This is a good reference figure for comparison questions where you infer how $U$ changes without recalculating from scratch. Source

• If the distance doubles, $U$ becomes half as large.

• If the distance triples, $U$ becomes one-third as large.

• If the distance decreases, the magnitude of $U$ increases.

A very common mistake is to use a square on the distance. For electric potential energy between two charges, the relationship is $1/r$, not $1/r^2$.

Sign of the charges

The sign of $U$ comes directly from the sign of the product $q_1q_2$.

A dipole-style visualization shows positive potential (purple) near the positive charge and negative potential (blue) near the negative charge, with equipotential contours drawn as black curves. This reinforces that electrostatic quantities are signed scalars and that opposite-sign charge configurations correspond to a qualitatively different (lower/negative) energy landscape than like-charge configurations. Source

• If both charges have the same sign, then $q_1q_2$ is positive, so $U$ is positive.

• If the charges have opposite signs, then $q_1q_2$ is negative, so $U$ is negative.

This means the sign of electric potential energy is not optional or decorative. It is a real part of the answer and should be kept throughout the calculation.

Using the equation correctly

When you use $U=\dfrac{kq_1q_2}{r}$, treat it as an equation for the entire two-charge system, not for just one object by itself. The answer represents the energy of the pair in that particular arrangement.

A reliable method is:

• write each charge with its correct sign

• convert all charges into coulombs

• convert the separation into meters

• substitute directly into the equation

• keep the sign from the product $q_1q_2$

• report the answer in joules

Unit conversion matters a great deal. In many AP Physics 2 Algebra problems, the charges are given in microcoulombs or nanocoulombs. If those are not converted to coulombs before substitution, the final answer will be incorrect by a large factor.

The separation $r$ must match the distance used by the model. In most standard problems, it is the straight-line separation between the two charged objects.

What the value of $U$ tells you

The numerical value of electric potential energy gives information about the electrical state of the pair of charges at that separation.

A positive value means the product $q_1q_2$ is positive. A negative value means the product is negative. A value with a larger absolute magnitude, such as $|U|=0.80\ J$ compared with $|U|=0.20\ J$, means the electrical interaction of the pair is more significant in that configuration.

It is also useful to compare cases with the same distance:

• same-sign charges give a positive value

• opposite-sign charges give a negative value

• if the charge magnitudes are the same in both cases, the magnitudes of $U$ are the same but the signs are opposite

Common errors to avoid

Students often lose points on this topic because of a small setup error rather than a physics misunderstanding.

Watch for these common mistakes:

• dropping the sign of one charge and calculating only a positive answer

• using charge values in microcoulombs without converting to coulombs

• using centimeters instead of meters for separation

• using $1/r^2$ instead of the correct $1/r$ dependence

• treating the answer as if it belongs to one charge alone instead of the two-charge system

• forgetting that a negative answer can be completely correct

Careful attention to signs, units, and the exact form of the equation is the key to success on this subsubtopic.