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By the end of this lesson, students should be able to:
Define electrical potential energy.
Compare gravitational potential energy and electrical potential energy.
Calculate the electric potential energy of charge configurations.
Understand the relationship between electric force and potential energy.
Determine how electric potential energy changes when charges move.
Solve AP Physics C problems involving electrical potential energy.
Electrical potential energy is the energy stored in a system of charged particles due to their positions relative to one another.
Just as an object raised above Earth possesses gravitational potential energy, charged objects possess electrical potential energy when they are separated in an electric field.
Electrical potential energy is represented by:
$$
U
$$
and is measured in joules (J).
In electrostatics, charges exert forces on one another.
Whenever forces act over a distance, energy considerations become useful.
Using energy methods often simplifies problems involving:
Electric fields
Electric potential
Capacitors
Charge motion
Near Earth’s surface:
$$
U_g = mgh
$$
where:
(m) = mass
(g) = gravitational field strength
(h) = height
Objects naturally move toward lower gravitational potential energy.
Similarly, electric charges tend to move toward configurations with lower electrical potential energy.
The electrical potential energy of two point charges is:
$$
U = k\frac{q_1q_2}{r}
$$
where:
(U) = electrical potential energy
(k) = Coulomb constant
(q_1) and (q_2) = charges
(r) = separation distance
The Coulomb constant is:
$$
k = 8.99\times10^9 ; N\cdot m^2/C^2
$$
This constant appears throughout electrostatics.
For two positive charges:
$$
q_1q_2 > 0
$$
For two negative charges:
$$
q_1q_2 > 0
$$
Therefore:
$$
U > 0
$$
Positive potential energy corresponds to repulsive interactions.
For opposite charges:
$$
q_1q_2 < 0
$$
Therefore:
$$
U < 0
$$
Negative potential energy corresponds to attractive interactions.
Positive potential energy means external work was required to assemble the charges.
Negative potential energy means energy is released when the charges come together.
Systems naturally move toward lower potential energy.
The electric force is a conservative force.
This means that work done by the electric force depends only on the initial and final positions.
As a result:
$$
W = -\Delta U
$$
where:
(W) = work done by the electric force
(\Delta U) = change in potential energy
If the electric force does positive work:
$$
W > 0
$$
then:
$$
\Delta U < 0
$$
Potential energy decreases.
If external work increases potential energy:
$$
\Delta U > 0
$$
then the electric force does negative work.
The change in potential energy is:
$$
\Delta U = U_f – U_i
$$
where:
(U_i) = initial potential energy
(U_f) = final potential energy
Using:
$$
U = k\frac{q_1q_2}{r}
$$
the change in potential energy becomes:
$$
\Delta Ukq_1q_2\left(\frac{1}{r_f}\frac{1}{r_i}\right)
$$
This equation frequently appears on AP exams.
When a charge is moved slowly so that kinetic energy remains constant:
$$
W_{ext}=\Delta U
$$
The external work equals the increase in potential energy.
Like charges repel.
To move them closer together requires positive external work.
Therefore:
$$
\Delta U > 0
$$
Potential energy increases.
Opposite charges attract.
As they move closer:
$$
\Delta U < 0
$$
Potential energy decreases.
Energy is released by the system.
For an isolated system:
$$
K_i + U_i=K_f + U_f
$$
where:
(K) = kinetic energy
(U) = potential energy
Often written as:
$$
\Delta K=-\Delta U
$$
A decrease in potential energy produces an increase in kinetic energy.
Two charges are separated by 0.50 m.
$$
q_1 = 2.0\times10^{-6} C
$$
$$
q_2 = 3.0\times10^{-6} C
$$
Find the electrical potential energy.
Use:
$$
U = k\frac{q_1q_2}{r}
$$
Substitute:
$$
U=(8.99\times10^9)\frac{(2.0\times10^{-6})(3.0\times10^{-6})}{0.50}
$$
$$
U = 0.108 J
$$
$$
U = 0.108 J
$$
The result is positive because both charges have the same sign.
Two charges are separated by 0.20 m.
$$
q_1 = 4.0\times10^{-6} C
$$
$$
q_2 = -6.0\times10^{-6} C
$$
Find the electrical potential energy.
$$
U=(8.99\times10^9)\frac{(4.0\times10^{-6})(-6.0\times10^{-6})}{0.20}
$$
$$
U = -1.08 J
$$
$$
U = -1.08 J
$$
The negative sign indicates an attractive interaction.
Potential energy diagrams are often used to analyze charge motion.
Key ideas:
Lower potential energy corresponds to greater stability.
Charges naturally move toward lower potential energy.
Steeper slopes correspond to stronger electric forces.
A stable equilibrium occurs at a minimum in the potential energy curve.
Small displacements produce restoring forces.
Confusing electric potential energy with electric potential.
Potential energy:
$$
U
$$
Electric potential:
$$
V
$$
These are different quantities.
Ignoring the sign of charges.
Remember:
$$
q_1q_2 > 0
\Rightarrow
U > 0
$$
$$
q_1q_2 < 0
\Rightarrow
U < 0
$$
Using distance squared.
The electrical potential energy equation contains:
$$
\frac{1}{r}
$$
not
$$
\frac{1}{r^2}
$$
The inverse-square relationship belongs to electric force and electric field.
Always determine:
Initial separation
Final separation
Initial potential energy
Final potential energy
before calculating work or energy changes.
Whenever charges move freely:
$$
K_i + U_i=K_f + U_f
$$
Energy methods are often easier than force-based approaches.
Electrical potential energy is stored energy associated with charge configurations.
$$
U = k\frac{q_1q_2}{r}
$$
Like charges have positive potential energy.
Opposite charges have negative potential energy.
Electric force is conservative.
$$
W = -\Delta U
$$
Potential energy decreases when the electric force does positive work.
Energy conservation relates kinetic and potential energy.
$$
K_i + U_i = K_f + U_f
$$
Understanding electrical potential energy is essential before studying electric potential, voltage, and capacitance.
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