Premed · Premed · Physics 2
Lecture 4: Electric Potential and Potential Energy
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Define electric potential energy and electric potential, and distinguish between the two
- Calculate the electric potential due to point charges and charge distributions
- Relate the electric field to the electric potential using the gradient relationship
- Draw and interpret equipotential surfaces for various charge configurations
- Apply conservation of energy to solve problems involving charged particles moving through potential differences
Lecture Content
I. Electric Potential Energy
Electric potential energy (U) is the energy stored in a system of charges due to their relative positions. For two point charges q1 and q2 separated by distance r, the potential energy is U = kq1 q2 / r. Unlike the force expression in Coulomb's law, no absolute value is taken here because the sign carries physical meaning. When U is positive (like charges), energy must be supplied to bring the charges together. When U is negative (unlike charges), energy is released as the charges come together. The reference point is chosen so that U = 0 when the charges are infinitely far apart.
For a system of more than two charges, the total potential energy is the sum over all unique pairs: U_total = sum over all pairs (i < j) of kq_i q_j / r_ij. The work done by the electric force is W_elec = -Delta U = -(U_final - U_initial). Because the electric force is conservative, this work is independent of the path taken between the initial and final configurations.
II. Electric Potential (Voltage)
Electric potential (V) is the electric potential energy per unit charge: V = U / q_0, where q_0 is a test charge. Equivalently, V at a point is the work done per unit charge by an external agent to bring a positive test charge from infinity to that point. The SI unit of potential is the volt (V), equal to 1 J/C.
The potential due to a single point charge Q at distance r is V = kQ / r. Unlike the electric field, potential is a scalar quantity with no direction, only a sign. Positive charges create positive potential, while negative charges create negative potential. The superposition principle for potential is especially convenient: V_total = V_1 + V_2 + V_3 + ..., an algebraic sum that is much simpler to evaluate than the vector addition required for fields.
It is the potential difference (Delta V = V_B - V_A), also called the "voltage," that is physically meaningful. This quantity is independent of the choice of reference point.
III. Relationship Between Electric Field and Potential
The potential difference between two points is related to the electric field by Delta V = V_B - V_A = -integral from A to B of E dot dl. This tells us that the electric field points from regions of high potential to regions of low potential.
For a uniform electric field, this relationship simplifies to Delta V = -E d, where d is the displacement parallel to E, giving E = -Delta V / d. More generally, the electric field is the negative gradient of the potential: E = -grad V = -(dV/dx x_hat + dV/dy y_hat + dV/dz z_hat). The field therefore points in the direction of the steepest decrease in potential.
This relationship also provides a useful unit: the electric field can be expressed in V/m, which is equivalent to N/C. A particularly convenient energy unit in atomic and nuclear physics is the electron volt (eV), defined as the energy gained by an electron accelerated through a potential difference of 1 V: 1 eV = 1.602 x 10^-19 J.
<image>A two-dimensional plot showing equipotential lines (dashed concentric circles at V = 10 V, 20 V, 30 V) around a positive point charge at the center, with electric field lines (solid arrows) radiating outward. Arrows labeled E are perpendicular to the equipotential lines everywhere. A path from point A to point B is drawn, with the integral of E dot dl indicated along the path. A callout box notes that E points from high V to low V, and that field lines are always perpendicular to equipotential surfaces.</image>
IV. Equipotential Surfaces
An equipotential surface is a surface on which the potential has the same value at every point. No work is done when a charge moves along an equipotential surface, since Delta V = 0 implies W = 0. Electric field lines are always perpendicular to equipotential surfaces, and equipotential surfaces never cross each other. Where equipotentials are closely spaced, the electric field is strong.
The shapes of equipotential surfaces depend on the charge configuration. For a point charge, they are concentric spheres centered on the charge. For a uniform field, they are parallel planes perpendicular to the field. For a dipole, they form complex three-dimensional surfaces. The surface of a conductor in electrostatic equilibrium is always an equipotential surface, and since E = 0 inside the conductor, the entire conductor is at the same potential.
V. Potential Due to Continuous Charge Distributions
The general approach for continuous distributions is V = integral of k dq / r. Because V is a scalar, this calculation is simpler than the corresponding field calculation, which requires vector components.
For a uniformly charged ring of total charge Q and radius a, the potential on the axis at distance x from the center is V = kQ / sqrt(x^2 + a^2). This result follows from the fact that all elements of charge on the ring are equidistant from any axial point.
For a uniformly charged disk with surface charge density sigma and radius R, the axial potential is V = (sigma / 2 epsilon_0)(sqrt(x^2 + R^2) - x).
For a uniformly charged sphere of total charge Q and radius R, the potential outside (r > R) is V = kQ / r, while inside (r < R) it is V = kQ(3R^2 - r^2) / (2R^3) for an insulating sphere. The potential is continuous at r = R. Once V is known everywhere, the electric field can be recovered by taking the negative gradient.
VI. Conservation of Energy with Electric Potential
For a charge q moving through a potential difference, conservation of energy gives Delta KE + Delta U = 0, or equivalently (1/2)mv_f^2 - (1/2)mv_i^2 = -q(V_f - V_i) = q(V_i - V_f). A positive charge released from rest accelerates from high potential to low potential, while a negative charge released from rest accelerates from low potential to high potential.
This principle has many practical applications. Particle accelerators use potential differences to give charged particles enormous kinetic energies. Cathode ray tubes accelerate electrons toward a phosphor screen. X-ray tubes accelerate electrons through potential differences of roughly 50-100 kV before they strike a metal target, producing X-rays.
<image>An energy diagram showing a positive charge q moving from point A (high potential V_A) to point B (low potential V_B) in a uniform electric field between parallel plates. The left side shows the physical setup with the charged plates and the particle's trajectory. The right side shows a bar chart comparing energies: at point A, the bar shows high potential energy U_A and zero kinetic energy; at point B, the bar shows lower potential energy U_B and higher kinetic energy KE_B, with total energy (U + KE) remaining constant. The equation (1/2)mv^2 = q(V_A - V_B) is displayed.</image>
