Introduction to Electric Potential

In the previous chapter, we looked at Electric Potential Energy (\(U_E\)). While potential energy is great for describing a system of charges, physicists often want to know about the "electrical environment" created by a single charge or a distribution of charges, regardless of what other particles might be placed there. This is where Electric Potential (\(V\)) comes in.

Think of it like a mountain: the height of the mountain at a certain spot is like the Electric Potential. It doesn't matter if a boulder or a pebble is sitting there; the height remains the same. However, the Potential Energy depends on the mass of the object sitting at that height. In our electrical world, Electric Potential is the "electrical height" of a point in space!

1. Defining Electric Potential

Electric Potential (\(V\)) is defined as the electric potential energy per unit charge. Mathematically, it looks like this:

\(V = \frac{U_E}{q}\)

Key Facts to Remember:

  • Unit: The unit for potential is the Volt (\(V\)), which is equal to one Joule per Coulomb (\(1 \text{ V} = 1 \text{ J/C}\)).
  • Scalar Quantity: Unlike the Electric Field, Electric Potential is a scalar. It does not have a direction! This makes the math much easier because you don't have to worry about x and y components.
  • The Zero Point: By convention, for isolated point charges, we set the electric potential to be zero at an infinite distance away (\(V = 0\) at \(r = \infty\)).
The Relationship with the Electric Field

The change in electric potential (\(\Delta V\)) between two points is related to the work done by the electric field as a charge moves. We calculate this using a line integral:

\(\Delta V = V_b - V_a = -\int_a^b \vec{E} \cdot d\vec{l}\)

Pro-tip: Notice the negative sign! The electric potential decreases as you move in the direction of the electric field lines. Think of field lines as "downhill" streams.

Key Takeaway: Electric potential describes the "potential-ness" of a location in space, measured in Volts. High potential is near positive charges; low potential is near negative charges.

2. Potential of Point Charges and Superposition

For a single point charge \(q\), the electric potential at a distance \(r\) away is given by:

\(V = \frac{kq}{r} = \frac{1}{4\pi\epsilon_0} \frac{q}{r}\)

What if there are multiple charges?

Because potential is a scalar, we use the Principle of Superposition. To find the total potential at a point, you simply add up the potentials from each individual charge. No vectors, no trigonometry—just simple addition!

\(V_{total} = \sum \frac{kq_i}{r_i}\)

Don't forget: You must include the sign of the charge. Positive charges create positive potential, and negative charges create negative potential.

3. Potential of Continuous Charge Distributions

When charge is spread out over an object, we use calculus. We break the object into tiny "pieces" of charge \(dq\) and integrate. The general formula is:

\(V = \int \frac{k \, dq}{r}\)

According to the AP Physics C syllabus, you are only expected to handle four specific geometries using calculus:

1. Thin Ring of Charge (on its axis)

For a ring of radius \(R\) and total charge \(Q\), the potential at a point \(x\) along the central axis is:

\(V = \frac{kQ}{\sqrt{x^2 + R^2}}\)

Why? Every little piece of charge \(dq\) on the ring is the same distance \(r = \sqrt{x^2 + R^2}\) from the point on the axis.

2. Semicircular Arc (at its center)

For an arc of radius \(R\) and total charge \(Q\), the potential at the center of the curvature is very simple:

\(V = \frac{kQ}{R}\)

3. Finite Wire or Line Charge

You may be asked to find the potential at a point collinear with the wire (on the same line) or on its perpendicular bisector. You will set up the integral \(dq = \lambda \, dx\) and integrate over the length of the wire.

4. Infinite Uniformly Charged Wire or Cylinder

Since an infinite wire has infinite charge, we cannot set \(V = 0\) at infinity. Instead, we use the integral of the electric field (\(E = \frac{\lambda}{2\pi\epsilon_0 r}\)):

\(V_b - V_a = -\int_a^b \frac{\lambda}{2\pi\epsilon_0 r} dr = -\frac{\lambda}{2\pi\epsilon_0} \ln\left(\frac{r_b}{r_a}\right)\)

Quick Review: When dealing with continuous distributions, identify \(dq\) in terms of charge density (\(\lambda\), \(\sigma\), or \(\rho\)) and integrate based on the distance \(r\) to your point of interest.

4. Finding the Electric Field from the Potential

If you know the potential \(V\) as a function of position, you can find the Electric Field (\(E\)) by taking the derivative. In one dimension:

\(E_x = -\frac{dV}{dx}\)

In three dimensions, the electric field is the negative gradient of the potential. This means the electric field points in the direction where the potential drops most sharply.

Common Mistake: Forgetting the negative sign! The electric field always points from high potential to low potential.

5. Equipotential Surfaces

An equipotential surface is a three-dimensional surface where every point on the surface has the exact same electric potential.

  • Perpendicularity: Electric field lines are always perpendicular to equipotential surfaces.
  • Work: No work is done moving a charge along an equipotential surface (\(W = -q\Delta V = 0\)).
  • Spacing: Where equipotential lines are crowded together, the electric field is stronger. Where they are spread apart, the field is weaker.

Example: Around a point charge, the equipotential surfaces are concentric spheres. For a uniform electric field (like inside a capacitor), the equipotential surfaces are parallel planes.

Summary Checklist

• Definition: \(V = U_E / q\) (Scalar, Volts).

• Point Charge: \(V = kq/r\).

• Finding V from E: \(\Delta V = -\int \vec{E} \cdot d\vec{l}\).

• Finding E from V: \(E = -dV/dr\).

• Equipotentials: Always perpendicular to \(\vec{E}\); no work done moving along them.

Don't worry if the integration seems tricky at first! Focus on setting up the expression for \(dq\) and \(r\). Once the integral is set up, the physics is mostly done, and the rest is just calculus practice.