Unit 9: Electric Potential

Electric Potential Energy

Definition and Concept

• Electric potential energy is the energy stored in a system due to the relative positions of charged particles. It arises from the electric force and represents the ability of the system to do work due to charge separation. This is analogous to gravitational potential energy, where energy depends on height — except here, energy depends on electric position in a field.
• For two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \), the electric potential energy is given by \( U = \frac{k q_1 q_2}{r} \), where \( k \) is Coulomb’s constant. This equation is derived from integrating the electric force over distance and assumes zero potential energy at infinite separation.
• The sign of \( U \) is important. If the charges are like-signed, \( U \) is positive, meaning energy is required to bring them closer (they repel). If the charges are opposite, \( U \) is negative, indicating energy is released as they come together (they attract). This sign convention is critical when applying conservation of energy.

Work and Energy Transformations

• Electric potential energy changes when work is done by or against an electric field. If an external force moves a charge against the field direction (e.g., moving a positive charge toward another positive charge), energy is stored in the system, and \( U \) increases. If the field moves the charge naturally, energy is released and \( U \) decreases.
• The work done by an electric field is equal to the negative change in potential energy: \[ W_{\text{field}} = -\Delta U = q\Delta V \] This relationship ties electric potential energy directly to electric potential, which simplifies many AP Physics C problems where forces are not explicitly calculated.
• In problems involving motion (like a proton accelerating from rest), energy conservation is often written as \[ \Delta K + \Delta U = 0 \] or \[ \frac{1}{2}mv^2 = q\Delta V \] This is especially useful when charges are released in uniform fields or between plates and need to be analyzed with no net external work.

Electric Potential (Voltage)

Definition and Conceptual Meaning

• Electric potential, often called voltage, is a scalar quantity that represents the electric potential energy per unit charge at a point in space. It tells us how much potential energy a positive test charge would have if placed at that point. The concept of voltage is useful because it simplifies electric interactions by focusing on position rather than force.
• Mathematically, electric potential is defined as \[ V = \frac{U}{q} \] where \( V \) is potential, \( U \) is electric potential energy, and \( q \) is the charge. This formula allows us to treat electric potential independently of any specific test charge, making it easier to map electric fields and solve for energy changes.
• Electric potential is a **relative** quantity — only differences in potential (\( \Delta V \)) matter physically. We often define \( V = 0 \) at infinity when dealing with point charges, or at the negative terminal of a battery in circuits. This freedom allows us to simplify calculations without changing the underlying physics.

Units and Interpretation

• The SI unit of electric potential is the volt (V), which is equal to one joule per coulomb: \[ 1 \, \text{V} = 1 \, \frac{\text{J}}{\text{C}} \] This unit expresses how much energy a 1-coulomb charge would gain or lose when moving through a potential difference of 1 volt.
• A positive potential means a positive test charge would gain potential energy at that location, while a negative potential means it would lose energy. For example, near a positive charge, the potential is high, and near a negative charge, the potential is low — regardless of whether a charge is actually placed there.
• Because electric potential is a scalar, potentials from multiple charges can be added algebraically, unlike electric fields which require vector addition. This makes voltage easier to compute in systems of multiple charges and especially useful in energy-based methods of problem solving.

Relationship Between Electric Field and Potential

Conceptual Link Between Field and Potential

• The electric field represents the force per unit charge, while electric potential represents the energy per unit charge. These two quantities are directly related: the electric field points in the direction of greatest decrease of electric potential. This means that a positive test charge naturally accelerates in the direction that lowers its potential energy.
• The electric field is strongest where the potential changes most rapidly with position. This idea mirrors the concept of a slope in calculus: a steep hill has a high rate of change in height, just as a strong field has a large change in voltage over a small distance. Understanding this gradient concept is essential for linking scalar potentials with vector fields.
• For example, in a uniform electric field, the field lines are straight and equally spaced, and the potential drops at a constant rate across distance. In contrast, near a point charge, the field decreases nonlinearly with distance, and so does the potential. This variation can be visualized with equipotential lines or graphs.

Mathematical Relationship

• The electric field is the negative spatial derivative of the electric potential: \[ E = -\frac{dV}{dx} \] This equation means that the electric field is the slope of the potential vs. position graph. The negative sign indicates that the field points in the direction of decreasing potential.
• In multiple dimensions, this relationship generalizes to a vector equation: \[ \vec{E} = -\nabla V \] where \( \nabla V \) is the gradient of the potential. This formalism is useful when calculating fields from known potentials, especially when symmetry or geometry makes the potential easier to determine.
• This derivative relationship also explains why electric fields are perpendicular to equipotential surfaces. Since potential doesn't change along an equipotential surface, the gradient (and thus the field) must be directed perpendicular to that surface. This helps visualize the direction of force on charges in diagrams.

Equipotential Surfaces

Definition and Properties

• Equipotential surfaces are imaginary surfaces where the electric potential is the same at every point. Since the potential is constant, a charge placed anywhere on the surface will have the same electric potential energy. No work is required to move a charge along an equipotential surface because there is no change in potential energy.
• These surfaces are always **perpendicular to electric field lines**. This follows from the fact that electric fields point in the direction of the greatest decrease in potential — and since there is no change in potential along an equipotential surface, the field must cross it at a right angle.
• The shape of equipotential surfaces depends on the field source. For a point charge, they are concentric spheres centered on the charge. For a uniform electric field, equipotentials are flat planes that are perpendicular to the field lines and equally spaced to reflect the uniform potential gradient.

Work and Energy on Equipotentials

• Because electric potential is constant along an equipotential surface, the **electric field does no work** when a charge moves along it. This means moving a charge around on the surface changes neither its speed nor its energy — motion is force-free in the tangential direction.
• In contrast, moving a charge between different equipotential surfaces requires work and results in a change in electric potential energy. The amount of work depends on the potential difference and the charge: \[ W = q \Delta V \] where a positive \( \Delta V \) implies work done against the field and an increase in potential energy.
• Equipotential diagrams can be used to visualize the direction and strength of electric fields. Closely spaced equipotentials imply a large change in potential over a small distance, meaning the electric field is strong in that region. Widely spaced surfaces indicate a weak field.

Calculating Potential for Point Charges

Electric Potential Due to a Single Point Charge

• The electric potential \( V \) at a distance \( r \) from a point charge \( q \) is given by the equation \[ V = \frac{kq}{r} \] where \( k = 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \). This formula assumes that the potential is zero at infinity, which is a standard convention in electrostatics.
• Electric potential is a **scalar**, so it has no direction and can be added algebraically. The sign of the potential depends on the sign of the charge: positive charges produce positive potential, while negative charges produce negative potential. This reflects whether a positive test charge would gain or lose potential energy at that location.
• The potential decreases with distance from the source charge, but unlike the electric field (which falls off as \( 1/r^2 \)), the potential falls off more slowly, as \( 1/r \). This slower falloff is important in systems where energy is more relevant than force, such as in energy conservation problems involving moving charges.

Electric Potential Due to Multiple Point Charges

• Because electric potential is a scalar, the total potential at any point due to multiple charges is the **algebraic sum** of the individual potentials from each charge. This is much simpler than calculating the net electric field, which requires vector addition.
• For example, to find the potential at a point due to three charges, use \[ V_{\text{net}} = \sum_i \frac{kq_i}{r_i} \] where each \( r_i \) is the distance from the \( i \)th charge to the point of interest. Be sure to include the sign of each charge to properly account for positive and negative contributions to potential.
• This method is frequently tested in AP Physics C problems, particularly in situations where charges are arranged symmetrically (such as on a square or triangle). Students should be able to determine both the total potential at a location and how it compares to potential at other points in space.

Electric Potential Due to Continuous Charge Distributions

Using Integration to Find Electric Potential

• When charge is spread continuously over an object, we must use calculus to compute the electric potential. Instead of summing discrete point charges, we treat the object as composed of infinitesimal charge elements \( dq \), each of which contributes a tiny amount of potential \( dV = \frac{k \, dq}{r} \) at the observation point.
• The total potential is found by integrating: \[ V = \int \frac{k \, dq}{r} \] where \( r \) is the distance from the charge element \( dq \) to the observation point. Because potential is a scalar, this integral is typically easier than the corresponding field calculation, which would require vector components.
• The key steps are to: (1) express \( dq \) in terms of the charge density (linear, surface, or volume), (2) write \( r \) as a function of position, and (3) determine limits of integration based on the geometry. Symmetry and substitution techniques are often required for solving the integral analytically.

Examples of Common Charge Distributions

• For a **ring of charge** with total charge \( Q \) and radius \( R \), the potential at a point on the axis (a distance \( x \) from the center) is \[ V = \frac{kQ}{\sqrt{x^2 + R^2}} \] This result comes from integrating over the ring, noting that all \( dq \) elements are the same distance \( r = \sqrt{x^2 + R^2} \) from the point.
• For an **infinite line of charge** with linear charge density \( \lambda \), the potential is not finite at infinity, so we choose a finite reference point (usually \( r_0 \)). The result for potential at distance \( r \) is \[ V(r) = -\frac{2k\lambda}{r} + C \] where \( C \) is chosen based on the reference location. This setup is also helpful when finding the field via differentiation: \( E = -\frac{dV}{dr} \).
• For a **uniformly charged disk**, the integration is more complex but still manageable due to radial symmetry. These problems often require setting up in polar coordinates and evaluating nested integrals over radius and angle. The resulting potential is useful in modeling parallel-plate capacitors and fields in planar geometries.

Potential Energy of a System of Charges

Energy from Interactions Between Charges

• The total electric potential energy of a system of charges arises from the work required to assemble the configuration from infinity. For a pair of point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \), the potential energy is given by \[ U = \frac{k q_1 q_2}{r} \] which reflects the energy stored due to their mutual interaction.
• The sign of the energy depends on the charge signs: like charges yield positive potential energy (they repel), while opposite charges yield negative potential energy (they attract). This matches the intuition that energy is required to bring like charges together, and energy is released when opposites attract.
• In multi-charge systems, the total potential energy is the sum of the potential energies of **every unique pair** of charges. For example, in a system with three charges \( q_1, q_2, q_3 \), the total energy is: \[ U_{\text{total}} = \frac{k q_1 q_2}{r_{12}} + \frac{k q_1 q_3}{r_{13}} + \frac{k q_2 q_3}{r_{23}} \] Each term represents the work needed to assemble that pair while holding the others fixed.

Energy Interpretation and Applications

• Potential energy in charge systems determines their stability. Systems with negative total potential energy are generally more stable because it would require external energy input to break them apart. This principle underlies the formation of molecules, ionic bonds, and atomic structures.
• Potential energy also plays a key role in conservation of energy. When a charge moves in an electric field, changes in its potential energy are converted into kinetic energy or vice versa. In calculations, this is often expressed as: \[ \Delta U = -q \Delta V \quad \text{or} \quad \frac{1}{2}mv^2 = q \Delta V \] allowing you to solve for speed, displacement, or voltage without using force or acceleration.
• This topic also links to electric potential diagrams and graphs. Regions with steep potential gradients indicate strong fields and rapid changes in energy. Understanding these energy maps helps students reason qualitatively about where charges want to move and how much energy is gained or lost.

Electric Potential Due to Point Charges and Uniform Fields

Point Charges

• The electric potential \( V \) due to a point charge \( q \) at a distance \( r \) is given by \[ V = \frac{kq}{r} \] where \( k \) is Coulomb’s constant. This formula assumes zero potential at infinity and shows that potential decreases with distance from the charge.
• Since potential is a scalar, multiple point charges can be handled by simply summing their individual contributions: \[ V_{\text{net}} = \sum \frac{kq_i}{r_i} \] This is far easier than calculating the vector sum of electric fields, making potential a powerful tool for analyzing multi-charge systems.

Uniform Electric Fields

• In a uniform electric field \( \vec{E} \), such as the field between two parallel plates, the potential difference between two points separated by a distance \( d \) is given by \[ \Delta V = -Ed \] where the negative sign indicates that potential decreases in the direction of the electric field.
• This linear relationship is useful for calculating the speed of charged particles accelerated across a known voltage. For example, if an electron moves from a region of low potential to high potential (against the field), it loses kinetic energy, and vice versa.

Electric Potential Due to Configurations of Charge

Superposition for Discrete Charges

• For systems with multiple discrete charges (like dipoles or triangle/square arrangements), the total electric potential at a point is the sum of the potentials due to each charge, regardless of direction. Because potential is a scalar, you can directly add positive and negative contributions without using vector components.
• For example, in an electric dipole, the potential at a point on the perpendicular bisector is zero due to symmetry. However, on the axis of the dipole, the net potential is nonzero and decreases rapidly with distance. These patterns are important for interpreting diagrams and energy behavior in dipole fields.

Continuous Charge Distributions

• For charge spread over lines, surfaces, or volumes, we integrate to find the total potential: \[ V = \int \frac{k \, dq}{r} \] where \( r \) is the distance from the charge element \( dq \) to the point of interest. The symmetry of the configuration usually determines whether this integral can be solved analytically.
• Examples include a uniformly charged ring (potential along axis), disk (potential along central axis), or infinite line (logarithmic potential). These systems often show up in AP problems involving motion of charges along symmetry axes.

Energy Conservation When Electric Charges Interact

Conservation of Mechanical Energy

• When a charged particle moves through an electric field, its total mechanical energy is conserved if no non-electric forces act. This means that the sum of its kinetic and electric potential energy remains constant: \[ K_i + U_i = K_f + U_f \] or \[ \frac{1}{2}mv^2 = q\Delta V \] for a charge moving in a uniform or conservative field.
• This relationship allows you to solve for final speed, displacement, or potential difference without needing to calculate force or acceleration. It's a powerful method for analyzing motion in capacitor plates, between point charges, or along field lines.

Applications to Interacting Charge Systems

• In multi-charge systems, potential energy is stored in the arrangement of the charges. Bringing two like charges closer requires positive work (energy input), while allowing opposite charges to approach each other releases energy. This behavior governs attraction, repulsion, and motion in charge interactions.
• In problems involving electric launchers, accelerators, or potential wells, energy conservation allows prediction of outcomes without solving complex equations of motion. For example, an electron accelerated across 100 V gains 100 eV of kinetic energy, which can be converted into velocity using basic physics.

Practice Problem 1: Electric Potential from Two Charges

Two point charges are placed along the x-axis: a charge of \( +3.0 \, \mu\text{C} \) is located at \( x = 0 \), and a charge of \( -2.0 \, \mu\text{C} \) is located at \( x = 4.0 \, \text{m} \). What is the electric potential at the point \( x = 2.0 \, \text{m} \)?

Step-by-Step Solution:

• Step 1: Use the electric potential formula for a point charge: \[ V = \frac{kq}{r} \] where \( r \) is the distance from the charge to the point of interest.
• Step 2: Find the distance from each charge to the point at \( x = 2.0 \, \text{m} \). The positive charge is 2.0 m away; the negative charge is also 2.0 m away.
• Step 3: Compute the potentials from each charge:
  \( V_1 = \frac{(8.99 \times 10^9)(+3.0 \times 10^{-6})}{2.0} = 13,485 \, \text{V} \)
  \( V_2 = \frac{(8.99 \times 10^9)(-2.0 \times 10^{-6})}{2.0} = -8,990 \, \text{V} \)
• Step 4: Add the potentials (since potential is scalar): \[ V_{\text{net}} = 13,485 - 8,990 = \boxed{4,495 \, \text{V}} \]

Practice Problem 2: Energy from Potential Difference

An electron starts from rest and is accelerated across a potential difference of 120 V. What speed does it gain after passing through the full voltage?

Step-by-Step Solution:

• Step 1: Use conservation of energy: \[ q\Delta V = \frac{1}{2}mv^2 \] where \( q = 1.6 \times 10^{-19} \, \text{C} \), \( \Delta V = 120 \, \text{V} \), and \( m = 9.11 \times 10^{-31} \, \text{kg} \).
• Step 2: Solve for \( v \): \[ v = \sqrt{\frac{2q\Delta V}{m}} = \sqrt{\frac{2(1.6 \times 10^{-19})(120)}{9.11 \times 10^{-31}}} \]
• Step 3: Plug in the numbers: \[ v = \sqrt{\frac{3.84 \times 10^{-17}}{9.11 \times 10^{-31}}} = \sqrt{4.21 \times 10^{13}} \approx \boxed{6.5 \times 10^6 \, \text{m/s}} \]

Common Misconceptions: Electric Potential

Misconception 1: Electric potential and electric potential energy are the same thing

• Many students confuse electric potential (\( V \)) with electric potential energy (\( U \)), thinking they are interchangeable. However, potential is energy per unit charge: \[ V = \frac{U}{q} \] It describes the condition of space, not the energy of a specific particle.
• Electric potential is independent of any test charge, while potential energy depends on both the charge and the potential: \( U = qV \). This distinction is especially important when using conservation of energy or calculating force indirectly from a potential field.

Misconception 2: Electric potential is a vector like electric field

• Electric potential is a scalar, not a vector. Students sometimes try to assign a direction to potential or add it using vector components, which is incorrect. Because it's scalar, potential from multiple charges simply adds algebraically, regardless of geometry.
• This makes calculating potential in multi-charge systems much easier than calculating electric fields, which require full vector addition. Always treat electric potential as a scalar quantity — direction only arises when computing electric field as the gradient of potential.

Misconception 3: Higher voltage always means stronger electric field

• Students often assume that high voltage automatically implies a strong electric field. In reality, the strength of the electric field depends on the **rate of change** of potential with distance, not the absolute voltage itself.
• Electric field is given by the spatial derivative: \[ E = -\frac{dV}{dx} \] So a large potential difference over a small distance means a strong field, but the same potential difference over a large distance means a weaker field.

Misconception 4: No potential means no electric field

• Just because the electric potential at a point is zero does **not** mean the electric field is zero there. Electric field depends on how the potential changes in space, not on the potential value itself.
• For example, at a point equidistant from two equal but opposite charges (a dipole midpoint), the potential is zero, but the electric field is strong because the potentials are changing rapidly in opposite directions. Students must look at potential *gradients*, not values, when reasoning about fields.

Misconception 5: Moving along an equipotential always means zero distance

• Students sometimes think that moving along an equipotential means staying at the same point. In fact, equipotentials are **entire surfaces or lines** where the potential is constant — you can move significant distances along one and still have zero change in voltage.
• This is why electric field lines are always perpendicular to equipotential surfaces — no work is done moving a charge tangentially, but work is done moving across them. Understanding this geometry is essential when interpreting diagrams or predicting motion of charges.