10.5 Electric Potential

Electric potential is the electric potential energy per unit charge at a point in space, and it adds as a scalar, so you can sum the contributions from each charge directly. Potential difference (voltage) tells you how much potential energy per coulomb changes when a charge moves between two points, and it connects to the electric field through the spacing of equipotential lines.

Because potential is a scalar, it is often faster to work with than electric field, and multiple-choice questions test how potential, field, and energy relate to each other. The same voltage idea carries over from electrostatics into circuits.

Potential Adds as a Scalar and Links to Field

• Electric potential is energy per charge ($V = \Delta U_E / q$), measured in volts, where 1 V = 1 J/C.
• Potentials from multiple point charges add as scalars: $V = \frac{1}{4\pi\varepsilon_0}\sum_i \frac{q_i}{r_i}$. No vector components needed.
• Potential can be positive or negative depending on the sign of the source charges, with the reference set at infinity.
• The electric field points toward decreasing potential, and its average magnitude is $|\vec{E}| = |\Delta V / \Delta r|$.
• Equipotential lines (isolines) are always perpendicular to field vectors, and no work is done moving a charge along an equipotential.
• Conductors in electrical contact redistribute charge until their surfaces sit at the same potential.

Electric Potential Due to Charged Objects

Electric potential represents the electric potential energy per unit charge at a specific point in space. This scalar quantity lets you analyze how charged particles interact with electric fields without tracking forces directly.

When multiple charges are present, use scalar superposition to find the total electric potential. Unlike electric fields, which are vectors, electric potentials add algebraically:

$V=\frac{1}{4 \pi \varepsilon_{0}} \sum_{i} \frac{q_{i}}{r_{i}}$

This equation shows that the electric potential:

• Increases in magnitude with larger source charges ($q_i$)
• Decreases with distance ($r_i$)
• Can be positive or negative depending on the sign of the charges

The electric potential difference between two points equals the change in electric potential energy per unit charge when moving a test charge between those points:

$\Delta V=\frac{\Delta U_{E}}{q}$

This potential difference can arise from:

• Spatial separation in an electric field
• Chemical processes that separate charge, as in a battery
• Other mechanisms that create charge separation

One useful property of conductors: when they are in electrical contact, electrons redistribute until all connected conducting surfaces reach the same electric potential. That is why connected conductors in electrostatic equilibrium share one uniform potential across their surfaces.

Relationship Between Electric Potential and Electric Field

Electric field and electric potential are closely linked. The average electric field between two points equals the electric potential difference divided by the distance between them:

$|\vec{E}|=\left|\frac{\Delta V}{\Delta r}\right|$

This relationship leads to several key ideas:

1. Electric field vectors point in the direction of decreasing potential (from high to low potential).
2. The magnitude of the electric field equals the rate at which potential changes with distance.

To picture these concepts, use two complementary representations:

• Electric field vector maps show the direction and strength of the field at various points.
• Equipotential lines (isolines) connect points of equal electric potential.

These representations follow specific rules:

• Equipotential lines are always perpendicular to electric field vectors.
• There is no component of the electric field along an equipotential; the field is entirely perpendicular to the isoline.
• No work is done when moving a charge along an equipotential line.
• The closer together equipotential lines are, the stronger the electric field.
• An isoline map can be built from an electric field vector map, and the reverse is also true.

Solving Potential Problems and Linking Representations

Problem Solving: For potential from several point charges, work charge by charge. Find each distance $r_i$, compute $kq_i/r_i$ keeping the sign of each charge, then add the results as scalars. There are no x and y components to break apart, which makes potential calculations less error-prone than field calculations if you stay careful with signs.

For uniform fields, use $|\vec{E}| = |\Delta V / \Delta r|$ to move between field and potential. Remember the field points toward lower potential, so a positive charge released from rest moves toward lower potential and a negative charge moves toward higher potential.

Free Response: This content is strong practice for the Translation Between Representations question. Be ready to:

• Sketch equipotential lines around a charged sphere or configuration of charges.
• Connect an equipotential map to an electric field vector map, using perpendicularity and isoline spacing.
• Tie those diagrams to energy bar charts for a charge released in the field, then explain in words why all the representations agree.

When you justify a claim, name the rule you are using, such as "field points toward decreasing potential" or "no work is done along an equipotential," instead of just stating the result.

Common Trap: Watch the difference between potential ($V$) and potential energy ($U_E$). Potential is per unit charge and exists in space whether or not a charge is there. Potential energy needs an actual charge in that location ($U_E = qV$). Mixing these up leads to unit errors and wrong signs.

Practice Problem 1: Electric Potential Due to Multiple Charges

Solution

Apply scalar superposition: find the contribution from each charge, then add them.

Step 1: Identify the position of each charge and the point where you need the potential.

• Charge 1: q₁ = +2.0 μC at (0, 0) m
• Charge 2: q₂ = -3.0 μC at (2, 0) m
• Charge 3: q₃ = +1.0 μC at (4, 0) m
• Point P is at (3, 0) m

Step 2: Calculate the distance from each charge to point P.

• r₁ = |3 - 0| = 3.0 m
• r₂ = |3 - 2| = 1.0 m
• r₃ = |3 - 4| = 1.0 m

Step 3: Calculate the electric potential due to each charge using V = k(q/r).

• V₁ = (9 × 10⁹ N·m²/C²)(2.0 × 10⁻⁶ C)/(3.0 m) = 6,000 V
• V₂ = (9 × 10⁹ N·m²/C²)(-3.0 × 10⁻⁶ C)/(1.0 m) = -27,000 V
• V₃ = (9 × 10⁹ N·m²/C²)(1.0 × 10⁻⁶ C)/(1.0 m) = 9,000 V

Step 4: Add the individual potentials to find the total potential at point P.

• V_total = V₁ + V₂ + V₃ = 6,000 V + (-27,000 V) + 9,000 V = -12,000 V

The electric potential at point (3.0 m, 0 m) is -12,000 V, or -12 kV.

Practice Problem 2: Equipotential Lines and Electric Field

Solution

In a uniform electric field, the electric potential varies only in the direction of the field. Since the field points in the positive x-direction, the potential changes only with the x-coordinate.

Step 1: Use the relationship between electric field and potential difference. Since E = -ΔV/Δx, rearrange to find ΔV = -E·Δx.

Step 2: Calculate the potential at the point (3.0 m, 4.0 m). The x-coordinate is 3.0 m, and the field is 200 N/C in the positive x-direction. V = V₀ - E·x = 0 - (200 N/C)(3.0 m) = -600 V

Step 3: Sketch the equipotential line. The equipotential line passing through (3.0 m, 4.0 m) is a vertical line at x = 3.0 m, since all points with x = 3.0 m share the same potential of -600 V, regardless of their y-coordinate. This line is perpendicular to the electric field vectors, which all point in the positive x-direction.

The electric potential at the point (3.0 m, 4.0 m) is -600 V, and the equipotential line is a vertical line at x = 3.0 m.

Common Misconceptions

• Misconception: Potential is the same as potential energy.
  Actually: Potential is not the same as potential energy. Potential ($V$) is energy per unit charge and exists at a point in space on its own. Potential energy ($U_E = qV$) requires a charge to actually be there.
• Misconception: Potential is a vector.
  Actually: Potential is a scalar, not a vector. You add potentials directly with signs included. Do not break them into components like you would with electric field.
• Misconception: Zero field means zero potential, and zero potential means zero field.
  Actually: Zero field does not mean zero potential, and zero potential does not mean zero field. For example, midway between two equal positive charges the field can be zero while the potential is not, and a point can have V = 0 while the field is nonzero.
• Misconception: The field points toward higher potential, not lower.
  Actually: The field points toward lower potential, not higher. A positive charge naturally moves toward lower potential; a negative charge moves toward higher potential.
• Misconception: Work is done along an equipotential.
  Actually: No work along an equipotential. Moving a charge along an equipotential line takes no work because the potential does not change, so kinetic energy from the field does not change either.
• Misconception: Farther-apart equipotential lines mean a stronger field.
  Actually: Closer equipotential lines mean a stronger field. Tight spacing signals a rapid change in potential over distance, which means a larger field magnitude.

Related AP Physics 2 Guides

• 10.1 Electric Charge and Electric Force
• 10.2 The Process of Charging
• 10.3 Electric Fields
• 10.6 Capacitors
• 10.7 Conservation of Electric Energy
• 10.4 Electric Potential Energy