Difficulty: Intermediate | Prerequisites: Basic calculus, vector notation, PHYS 211 (mechanics)
This material is the backbone of PHYS 212. Electrostatics describes how charges create electric fields and potentials, and Gauss's Law gives you a shortcut for computing those fields whenever symmetry cooperates. Capacitors store energy in electric fields and show up in every circuit topic later in the course. If you are comfortable with Coulomb's law, field superposition, and the idea that voltage is energy per charge, the rest of the semester builds on that foundation.
Charges produce electric fields; Gauss's Law lets you find those fields quickly when the geometry is spherical, planar, or cylindrical. Conductors in equilibrium have zero internal field and all excess charge on the surface. Capacitors store energy as (1/2)CV^2, and inserting a dielectric multiplies capacitance by the dielectric constant.
Electric field (E)
The force per unit charge at a point in space, measured in N/C (equivalently V/m). Think of it as the "push" a tiny positive test charge would feel if you placed it there.
Coulomb's law
The force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them: F = kq1q2/r^2. In simple terms, double the distance and the force drops to one quarter.
Gauss's Law
The net electric flux through any closed surface equals the enclosed charge divided by the permittivity of free space. In simple terms, it counts how much field "flows out" of a surface, and that total depends only on the charge inside.
Electric flux
The integral of E dot dA over a surface. Think of it as the number of field lines passing through that surface.
Capacitance (C)
The ratio of stored charge to voltage across a capacitor: C = Q/V, measured in farads (F). In simple terms, a bigger capacitance means the device holds more charge for the same voltage.
Dielectric constant (kappa)
A dimensionless number (always >= 1) describing how much an insulating material reduces the electric field between capacitor plates. Inserting a dielectric with kappa = 3 triples the capacitance.
Electric potential (V)
The electric potential energy per unit charge at a point, measured in volts. In simple terms, voltage tells you how much energy a charge gains or loses moving from one place to another.
Equipotential surface
A surface on which every point has the same electric potential. No work is done moving a charge along an equipotential.
Electrostatic equilibrium
The condition in which all charges in a conductor have stopped moving. The internal electric field is zero, all excess charge sits on the surface, and the surface is an equipotential.
Outside the sphere (r > R): the sphere behaves exactly like a point charge at its centre. E = kQ/r^2, where Q is the total charge and r is the distance from the centre.
Inside the sphere (r < R): only the charge enclosed within radius r contributes. Because charge is uniform, the enclosed charge scales as (r/R)^3 times Q. Gauss's Law on a spherical surface of radius r gives E = kQr/R^3.
The field increases linearly from zero at the centre to its maximum at the surface, then falls off as 1/r^2 outside.
Gauss's Law is always true, but it is only useful for computing E when symmetry lets you pull E out of the flux integral.
Spherical symmetry: point charges, uniformly charged spheres and shells. Use a concentric spherical Gaussian surface.
Cylindrical symmetry: infinite line charges, infinite uniformly charged cylinders. Use a coaxial cylindrical Gaussian surface.
Planar symmetry: infinite uniformly charged planes. Use a Gaussian "pillbox" straddling the surface.
A charged cube has no useful symmetry for Gauss's Law. You can still apply the law, but you cannot solve for E from it alone.
The electric field inside the bulk of a conductor is zero.
All excess charge resides on the surface.
The electric field at the surface is perpendicular to the surface (not parallel).
The entire conductor, surface included, is at the same potential (equipotential volume).
These properties hold regardless of the conductor's shape.
A parallel-plate capacitor has capacitance C = epsilon_0 * A / d, where A is the plate area and d is the separation.
Inserting a dielectric with constant kappa multiplies the capacitance: C' = kappa * C.
Battery connected (constant V): when the dielectric goes in, C increases but V stays fixed, so the charge on the plates increases (Q = CV). The stored energy U = (1/2)CV^2 also increases by a factor of kappa.
Battery disconnected (constant Q): when the dielectric goes in, C increases but Q is fixed, so V drops. The stored energy U = Q^2/(2C) decreases by a factor of kappa.
Mixing up these two scenarios is one of the most common exam mistakes.
The electric potential at a distance r from a point charge q is V = kq/r (taking V = 0 at infinity).
For multiple point charges, the potential at a point is the algebraic sum of the individual potentials (superposition, and potential is a scalar, so no vector addition needed).
The work done by an external agent to bring a charge Q from infinity to a point where the potential is V is W_ext = QV.
Sign matters: bringing a negative charge toward positive charges means the external agent does negative work (the field does positive work, pulling the charge in).
\vec{F} = k \frac{q_1 q_2}{r^2} \hat{r} \quad \text{(Coulomb's law)}\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} \quad \text{(Gauss's law)}E_{\text{inside sphere}} = \frac{kQr}{R^3} \quad (r < R) \qquad E_{\text{outside}} = \frac{kQ}{r^2} \quad (r > R)C = \frac{\varepsilon_0 A}{d} \quad \text{(parallel-plate capacitor)}C' = \kappa C \quad \text{(with dielectric)}U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV \quad \text{(energy stored)}V = \frac{kq}{r} \quad \text{(potential from a point charge)}W_{\text{ext}} = q \Delta V = q(V_f - V_i) \quad \text{(work by external agent)}Constants: k = 8.99 x 10^9 N m^2/C^2, epsilon_0 = 8.85 x 10^-12 F/m.
Capacitors with dielectrics are everywhere: the ceramic capacitors on a circuit board use high-kappa materials to pack more capacitance into a small space. Gauss's Law is how engineers model the field inside coaxial cables. The conductor-equilibrium rules explain why a Faraday cage blocks external electric fields.
Students often think Gauss's Law only applies to symmetric charge distributions. It is always true; symmetry just determines whether it is useful for computing E.
Students confuse the battery-connected and battery-disconnected cases for dielectrics. If the battery stays connected, voltage is constant and energy increases. If the battery is disconnected, charge is constant and energy decreases.
Students sometimes think the electric field inside a charged conductor is nonzero because there is charge present. The charges arrange themselves on the surface precisely so the interior field cancels to zero.
Students forget that electric potential is a scalar, not a vector. You sum potentials algebraically, not with vector components.
⚠️ Gauss's Law problems almost always test whether you can identify the right symmetry and Gaussian surface. If the problem says "uniformly charged sphere" or "infinite plane," Gauss's Law is the expected method.
⚠️ Dielectric problems will specify whether the battery remains connected or is disconnected. Read that detail first, because it determines which quantity stays constant.
⚠️ Work-done-by-external-agent problems require careful sign handling. The external agent's work equals the change in potential energy, which is qDeltaV. If q is negative and V is positive, the work is negative.
⚠️ Conductor equilibrium is a favourite for multi-select questions: all four properties (zero internal field, surface charge, perpendicular surface field, equipotential) tend to appear as options.
True or false: the electric field inside a uniformly charged solid insulating sphere is zero.
False. It is zero only at the very centre. The field increases linearly with r inside the sphere.
Fill in the blank: inserting a dielectric with kappa = 3 into a capacitor connected to a battery multiplies the stored energy by ___.
True or false: the electric field at the surface of a conductor in equilibrium is parallel to the surface.
False. It is perpendicular to the surface.
True or false: Gauss's Law can be used to find E for a charged cube.
False (in practice). Gauss's Law is always valid, but a cube lacks the symmetry needed to solve for E.
Q: A solid spherical insulator of radius R = 2.5 cm carries total charge Q = 8 microcoulombs uniformly distributed. What is the electric field at r = 5 cm from the centre?
A: Since r > R, treat it as a point charge. E = kQ/r^2 = (8.99 x 10^9)(8 x 10^-6)/(0.05)^2 = 2.88 x 10^7 N/C.
Q: A parallel-plate capacitor with area A and separation d is connected to a battery of voltage V. A dielectric with kappa = 3 is inserted while the battery stays connected. How does the stored energy change?
A: C becomes 3C. Since V is constant, U = (1/2)CV^2 increases by a factor of 3.
Q: Two charges +q are at (0, a) and (0, -a). What work does an external agent do to bring charge -Q from infinity to the origin?
A: The potential at the origin due to each +q charge is kq/a. Total potential V = 2kq/a. Work = (-Q)(2kq/a) = -2kqQ/a. The external agent does negative work (the field pulls the negative charge in).
Q: Name the four properties of a conductor in electrostatic equilibrium.
A: (1) E = 0 inside the bulk, (2) all excess charge is on the surface, (3) E at the surface is perpendicular to the surface, (4) the entire conductor is an equipotential volume.
Q: Which charge distributions allow Gauss's Law to be used to solve for E easily?
A: Those with spherical, cylindrical, or planar symmetry: point charges, uniformly charged spheres/shells, infinite lines/cylinders, and infinite planes. A charged cube does not qualify.
This material connects directly to DC circuits: capacitance appears in RC circuits, and the energy stored in a capacitor is what drives current when the battery is removed. Gauss's Law reappears in magnetism as a constraint (magnetic flux through any closed surface is zero), and the conductor-equilibrium ideas extend to shielding in electromagnetic interference problems.
Coulomb's law, electric field, Gauss's Law, Gaussian surface, electric flux, capacitance, parallel-plate capacitor, dielectric, dielectric constant, kappa, permittivity, epsilon naught, electric potential, voltage, equipotential, conductor equilibrium, Faraday cage, charge distribution, spherical symmetry, cylindrical symmetry, planar symmetry, superposition principle, PHYS 212, electricity and magnetism, E&M final review