Premed · Premed · Physics 2
Lecture 2: Electric Fields
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Define the electric field and explain the concept of a field as a mediator of force
- Calculate the electric field due to a point charge and multiple point charges
- Draw and interpret electric field lines for various charge configurations
- Calculate the electric field due to continuous charge distributions (line, ring, disk)
- Describe the motion of a charged particle in a uniform electric field
Lecture Content
I. The Electric Field Concept
The electric field is a vector field that exists in the space surrounding any electric charge. It is defined as the force per unit positive test charge placed at a given point: E = F / q_0, where q_0 is a small positive test charge. The test charge must be small enough that it does not disturb the source charges that create the field. The SI units of the electric field are newtons per coulomb (N/C), which are equivalent to volts per meter (V/m).
A crucial insight is that the field is a property of space itself, created by source charges. It exists whether or not a test charge is actually present to experience it. The force on any charge q placed in an existing electric field is simply F = qE. Positive charges experience a force in the direction of E, while negative charges experience a force opposite to E. The field concept elegantly replaces the notion of "action at a distance" with a local interaction: the source charge creates a field, and the field exerts a force on any other charge that enters it.
II. Electric Field of a Point Charge
For a single point charge Q located at the origin, the electric field at a distance r is E = kQ / r^2 * r_hat, directed radially outward for a positive charge Q. The magnitude is E = k|Q| / r^2. The field points away from positive charges and toward negative charges, and it has spherical symmetry, meaning that its magnitude is the same at all points equidistant from Q.
The superposition principle applies to electric fields just as it does to forces. The net electric field at any point due to multiple source charges is the vector sum of the individual fields: E_net = E_1 + E_2 + E_3 + ... Each contribution is calculated independently, and then the results are added as vectors.
III. Electric Field Lines
Electric field lines provide a powerful visual representation of the electric field. Several rules govern how they are drawn. Lines begin on positive charges and end on negative charges. They never cross one another, because the field has a unique direction at every point. The density of lines, measured as the number of lines per unit area perpendicular to them, is proportional to the field magnitude, so lines are closer together where the field is stronger. At any point, the direction of the field is tangent to the field line passing through that point.
Some common configurations are worth memorizing. A single positive charge produces field lines that radiate outward uniformly in all directions. A single negative charge produces lines that converge inward. An electric dipole generates lines that curve gracefully from the positive charge to the negative charge. Two like charges produce a pattern in which the field lines repel each other, with a null point between the charges where the field is zero. A parallel plate capacitor produces uniform, parallel field lines between the plates when edge effects are ignored.
<image>Four panels showing electric field line patterns. Panel A: Radial field lines pointing outward from a single positive point charge. Panel B: Radial field lines pointing inward toward a single negative point charge. Panel C: Field lines of an electric dipole — lines emerge from the positive charge, curve through space, and terminate on the negative charge, with a characteristic "figure-eight" pattern. Panel D: Two equal positive charges with field lines repelling each other, showing a point of zero field exactly halfway between them.</image>
IV. Electric Field of Continuous Charge Distributions
When charge is spread continuously over a region rather than concentrated at discrete points, the summation over individual contributions becomes an integral. The field from a small element of charge dq is dE = k dq / r^2 * r_hat, and the total field is obtained by integrating over the entire distribution: E = integral of dE.
To describe how charge is distributed, three charge densities are commonly used. The linear charge density lambda = Q/L (in C/m) describes charge along a line. The surface charge density sigma = Q/A (in C/m^2) describes charge on a surface. The volume charge density rho = Q/V (in C/m^3) describes charge filling a volume.
For an infinite line of charge with linear charge density lambda, the electric field at a perpendicular distance r from the line is E = 2k lambda / r = lambda / (2 pi epsilon_0 r). The field is directed radially outward from the line for positive lambda and falls off as 1/r, rather than the 1/r^2 dependence of a point charge.
For a ring of charge with total charge Q and radius a, the field at a point on the axis a distance x from the center is E_x = kQx / (x^2 + a^2)^(3/2). Only the axial component of the field survives, because symmetry ensures that the perpendicular components from opposite sides of the ring cancel exactly. At the center of the ring (x = 0), the field is zero, and far from the ring (x >> a), the field approaches kQ/x^2, recovering the point-charge result.
For a uniformly charged disk with surface charge density sigma and radius R, the axial field at distance x is E_x = (sigma / 2 epsilon_0) [1 - x / sqrt(x^2 + R^2)]. In the limit where R approaches infinity, this becomes E = sigma / (2 epsilon_0), a result that is constant and independent of distance. This uniform-field result is the foundation of the parallel plate capacitor.
<image>Panel A: A thin ring of charge with radius a centered at the origin in the y-z plane, with a point P on the x-axis at distance x from the center. A small element dq on the ring is shown, with the vector dE at point P decomposed into axial (dE_x) and perpendicular (dE_perp) components. Arrows indicate that perpendicular components from opposite sides of the ring cancel. Panel B: A uniformly charged disk of radius R, showing the disk as composed of many concentric rings, with the resultant electric field vector along the axis pointing away from the disk.</image>
V. Motion of Charged Particles in Electric Fields
A charge q placed in a uniform electric field E experiences a constant force F = qE. By Newton's second law, the resulting acceleration is a = F/m = qE/m. The motion is directly analogous to projectile motion in a gravitational field.
When the charge moves parallel to the field, it accelerates or decelerates uniformly. A positive charge accelerates in the direction of E, while a negative charge accelerates opposite to E.
When a charge enters a uniform field perpendicularly, such as between two parallel plates, its horizontal motion remains at uniform velocity because there is no force component in that direction. Its vertical motion, however, experiences uniform acceleration from the constant electric force. The resulting trajectory is parabolic, exactly like a projectile under gravity. This principle underlies the operation of cathode ray tubes, ink-jet printers, and electrostatic deflection systems.
<image>A diagram showing a charged particle (positive) entering horizontally between two parallel plates that create a uniform vertical electric field E pointing downward (from positive plate on top to negative plate on bottom). The particle's trajectory curves upward (toward the positive plate) following a parabolic path. The initial horizontal velocity v_0 is labeled, the vertical acceleration a = qE/m is indicated with an arrow, and the parabolic trajectory is drawn with a dashed curve. The deflection angle theta at the exit is labeled.</image>
VI. Electric Dipoles in Electric Fields
An electric dipole consists of two equal and opposite charges +q and -q separated by a distance d. The dipole moment is defined as p = qd, a vector pointing from -q to +q, measured in coulomb-meters (C m).
In a uniform electric field, the net force on the dipole is zero because the forces on +q and -q are equal in magnitude and opposite in direction. However, the dipole experiences a net torque given by tau = p x E, with magnitude tau = pE sin(theta), where theta is the angle between the dipole moment and the field. This torque tends to align the dipole with the field, and the configuration with p parallel to E is a stable equilibrium.
The potential energy of a dipole in a uniform field is U = -p dot E = -pE cos(theta). The energy is minimized when the dipole is aligned with the field (theta = 0) and maximized when it is anti-aligned (theta = pi).
In a non-uniform field, the dipole also experiences a net force directed toward regions of stronger field. This principle governs the behavior of polar molecules such as water in external electric fields and underlies the operation of microwave ovens, where the oscillating electric field causes water molecules to rotate rapidly, generating thermal energy.
