AP Physics 2: Electric Potential

Electric potential, often called voltage, is the invisible scaffold that shapes how charged particles move and store energy in everything from computer chips to biological cells. Unlike force, which tells you how a charge will be pushed, potential tells you about the energy landscape it navigates. Mastering this concept allows you to predict circuit behavior, understand nerve impulses, and analyze everything from capacitors to plasma physics.

Electric Potential Energy and Defining Voltage

To understand electric potential, you must first grasp electric potential energy ($U_e$). When you hold a book up against gravity, you give it gravitational potential energy; similarly, when you push a charge against an electric force, you store electric potential energy in the system. This energy depends on the configuration of charges.

The electric potential ($V$) at a point in space is defined as the electric potential energy per unit charge. If a test charge $q$ placed at a point has potential energy $U_e$, the electric potential at that point is:

$$V=\frac{U_e}{q}$$

The units are joules per coulomb, which are called volts (V). This definition is crucial: voltage is a property of the location in an electric field, not of the test charge you put there. A high potential point means a positive test charge placed there would have high potential energy. A key consequence is that the work done by the electric field when a charge moves between two points is directly related to the potential difference: $W_{field}=-q\Delta V$.

Electric Potential Due to a Point Charge

For the specific and foundational case of a single point charge $Q$, we can derive a formula for the potential it creates in the space around it. The potential at a distance $r$ from the point charge is given by:

$$V=\frac{kQ}{r}$$

where $k$ is Coulomb's constant ($8.99\times 10^9{\text{ Ncdotpm}}^2/{\text{C}}^2$). Notice this formula assumes the potential is zero at infinity ($r\to \infty$), which is the standard reference point. This is a scalar quantity, which simplifies calculations enormously compared to the vector electric field. The sign of the potential is determined by the sign of the source charge $Q$: positive charges create positive potentials, and negative charges create negative potentials, with the magnitude decreasing as $1/r$.

Example Calculation: What is the potential 0.5 meters from a +2 nC point charge?

1. Convert: $Q=2\times 10^{-9}\text{ C}$.
2. Apply the formula: $V=\frac{(8.99\times 10^9)(2\times 10^{-9})}{0.5}$.
3. Calculate: $V=\frac{17.98}{0.5}=35.96\text{ V}$.

The Superposition Principle for Multiple Charges

Because electric potential is a scalar quantity, the superposition principle is applied through simple algebraic addition. The total electric potential at a given point due to a collection of point charges is the sum of the potentials due to each individual charge.

$$V_{total}=V_1+V_2+V_3+...=\sum _i\frac{k{q}_i}{r_i}$$

You simply calculate the potential from each charge (mindful of its sign and distance from the point of interest) and add them together. This is significantly easier than vector addition for electric fields. This principle allows you to model the potential from complex charge distributions, such as dipoles or arrays of charges.

Worked Example: Two point charges, $q_1=+3\text{ nC}$ and $q_2=-5\text{ nC}$, are placed on the x-axis at $x=0$ m and $x=0.4$ m, respectively. Find the potential at point P located at $x=0.1$ m.

1. Distance from P to $q_1$: $r_1=0.1\text{ m}$.

Potential from $q_1$: $V_1=k(3\times 10^{-9})/0.1=(8.99\times 10^9)(3\times 10^{-8})=269.7\text{ V}$.

1. Distance from P to $q_2$: $r_2=0.3\text{ m}$.

Potential from $q_2$: $V_2=k(-5\times 10^{-9})/0.3=(8.99\times 10^9)(-1.667\times 10^{-8})\approx -149.8\text{ V}$.

1. Apply superposition: $V_{total}=V_1+V_2=269.7\text{ V}+(-149.8\text{ V})=119.9\text{ V}$.

Equipotential Surfaces and Their Properties

An equipotential surface is a three-dimensional surface on which the electric potential $V$ is the same at every point. Think of it like a contour line on a topographic map, where every point on that line is at the same elevation. For a single point charge, equipotential surfaces are concentric spheres. For a uniform electric field, they are planes perpendicular to the field lines.

Two critical rules govern equipotential surfaces:

1. No work is required to move a charge at constant speed along an equipotential surface. Since $\Delta V=0$, the work done by the electric field is zero.
2. Electric field lines are always perpendicular to equipotential surfaces. The field points in the direction of the steepest decrease in potential. The closer together the equipotential surfaces are, the stronger the electric field magnitude ($E=|\Delta V/\Delta d|$ in a uniform field).

Relating Equipotentials to Electric Fields

The relationship between $V$ and $\vec{E}$ is foundational. While $\vec{E}$ is a vector telling you the force per charge, $V$ is a scalar telling you the energy per charge. They are two complementary descriptions of the same underlying electric influence. Mathematically, the electric field component in any direction is the negative rate of change of the potential with distance in that direction. In calculus terms, $\vec{E}=-\nabla V$.

For the AP Physics 2 exam, the most practical takeaway is this: The electric field points from high potential to low potential, and its magnitude is related to how quickly the potential changes. If you map equipotential lines that are equally spaced in voltage, the regions where the lines are packed tightly together indicate a strong electric field. This visual link is powerful for analyzing problems involving conductors (which are equipotential objects) and complex field patterns.

Common Pitfalls

1. Confusing Potential with Potential Energy: Remember, potential ($V$) is "energy per charge," a property of space. Potential energy ($U_e$) is "energy of a specific charge" at a specific location. They are related by $U_e=qV$, but they are not interchangeable. A point can have a high potential, but if you place a negative charge there, it will have low (or negative) potential energy.
2. Sign Errors with Point Charges: The formula $V=kq/r$ inherently includes the sign of the source charge $q$. A common mistake is to take the absolute value and then try to assign a sign later. Always carry the sign of the charge through the calculation, especially when using the superposition principle.
3. Misinterpreting Equipotential Surfaces: It is incorrect to think the electric field is zero along an equipotential surface. The field is perpendicular to it. Zero work is done along the surface precisely because the force (from $\vec{E}$) is perpendicular to the displacement along the surface.
4. Assuming $\vec{E}=0$ where $V=0$: A point of zero potential does not imply a zero electric field. For example, at the midpoint of an electric dipole, the potentials from the equal and opposite charges cancel to zero, but the electric fields (vectors) add to a non-zero result pointing from the positive to the negative charge.

Summary

• Electric potential ($V$) is electric potential energy per unit charge, measured in volts. It is a scalar property of a point in space.
• For a point charge $Q$, the potential at a distance $r$ is $V=kQ/r$, with the sign of $Q$ included.
• For multiple charges, find the total potential by scalar addition: $V_{total}=\sum (k{q}_i/r_i)$. This is simpler than vector addition for electric fields.
• Equipotential surfaces connect points of equal potential. No work is done moving a charge along them, and they are always perpendicular to electric field lines.
• The electric field points from high to low potential, and its strength is proportional to how rapidly the potential changes with position ($E\approx |\Delta V/\Delta d|$).