wunder beta📘 How do fields carry force and energy?
Intermediate survey of electric and magnetic fields at 100 level
What you’ll learn
- Electric Charge and Coulomb ForceDefine charge quantization and derive the vector form of Coulomb's law.Charge exists in discrete multiples of e. The force between point charges follows an inverse-square law whose direction lies along the line joining the charges. Vector superposition extends the law to multiple charges.
- The Electric Field ConceptDefine the electric field as force per unit charge and compute it for point-charge distributions.The field is a vector function of position. For a point charge it points radially and falls as one over r squared. Superposition gives the net field from any static charge arrangement.
- Gauss's Law and SymmetryState Gauss's law and apply it to symmetric charge distributions.Electric flux through any closed surface equals enclosed charge divided by epsilon zero. High symmetry lets the flux integral collapse to a simple product of E and area.
- Electric Potential and Potential EnergyDefine electric potential and relate it to both field and stored energy.Potential difference equals the negative line integral of E. Equipotential surfaces are perpendicular to field lines. Potential energy for two point charges is k q1 q2 over r.
- Conductors and CapacitanceExplain induced charges on conductors and derive capacitance for parallel-plate geometry.Free charges rearrange until the internal field vanishes. Capacitance equals charge stored per volt applied. For parallel plates it depends only on area, separation, and the permittivity of the gap.
- Magnetic Field from Moving ChargesDefine the magnetic field and introduce the Biot-Savart law.Moving charges generate B fields. The Biot-Savart expression gives dB from each current element. Direction follows the right-hand rule.
- Ampere's Law and Ampèrian LoopsApply Ampere's law to infinite straight wires and solenoids.Ampere's law is the magnetic counterpart of Gauss's law. Cylindrical symmetry around a wire fixes B at mu zero I over two pi r. Inside a long solenoid the field is uniform and equal to mu zero n I.
- Faraday's Law of InductionState Faraday's law and calculate induced emf in simple circuits.Changing magnetic flux induces an electric field whose line integral around a closed path equals the negative rate of change of flux. Motional emf and transformer emf are two realizations of the same principle.
- Displacement Current and Maxwell's CorrectionIntroduce the displacement current and write the four Maxwell equations in integral form.The displacement current density equals epsilon zero times the time derivative of E. With this addition the equations become fully consistent for time-varying fields.
- Electromagnetic Waves in VacuumDerive the wave equation from Maxwell's equations and obtain the speed of light.The coupled curl equations produce second-order wave equations for both E and B. Plane-wave solutions travel at one over square root of mu zero epsilon zero. E and B are perpendicular to each other and to the propagation direction.
- Energy Density and Poynting VectorDefine electromagnetic energy density and the Poynting vector for power flow.Electric energy density is one-half epsilon zero E squared. Magnetic energy density is B squared over two mu zero. Their sum travels with the wave at the Poynting vector S equals E cross B over mu zero.
- Fields at the Boundary and RadiationState boundary conditions and sketch the radiation fields of an accelerating charge.Tangential E and normal B are continuous across an interface without surface current. Accelerating charges produce radiation whose power pattern and polarization follow from the Liénard-Wiechert fields.
Questions this course answers
Two protons are fixed 1 nm apart. Which change doubles the magnitude of their mutual repulsion?
An alpha particle carries charge +2e while a proton carries +e, so the product q1 q2 doubles and the force doubles. Distance, sign, and orientation affect the force differently.
A +2 µC charge sits at the origin. What is the electric field vector 0.5 m away along the positive x-axis?
E = kQ/r² directed radially outward for positive Q, so at x = 0.5 m the field points in the +x direction with magnitude 9 × 10⁹ × 2 × 10⁻⁶ / (0.5)² = 7.2 × 10⁴ N/C.
A spherical shell carries total charge Q. Which Gaussian surface lets you conclude immediately that E = 0 for r < R?
Only the concentric sphere matches the symmetry so that E is constant and the flux is exactly 4πr²E; Gauss’s law then forces E = 0 when Q_enc = 0.
A test charge +q is moved from point A to point B along two different paths in a static electric field. The work done by the field is 12 μJ along path 1 and W along path 2. What is W?
Because the line integral of E around any closed loop is zero, the work done by the field between two points is strictly path-independent.
A conducting sphere carries net charge +Q. A student places a Gaussian surface inside the metal. What is the electric flux through that surface?
In electrostatic equilibrium the field inside the conducting material is zero, so the flux through any Gaussian surface lying entirely inside the conductor must be zero by Gauss’s law.
A proton moves parallel to a long straight wire carrying current I. In which direction is the magnetic force on the proton?
The B field circles the wire; the proton’s velocity is parallel to the wire, so v cross B points radially inward or outward depending on current direction, producing an attractive force when currents are parallel.
Grounded in trusted sources
- Massachusetts Institute of Technology
- National Institute of Standards and Technology
- American Physical Society
- David J. Griffiths, Introduction to Electrodynamics — fields, potentials, Maxwell
- Hugh D. Young and Roger A. Freedman, University Physics — E&M chapters
- Edward M. Purcell and David J. Morin, Electricity and Magnetism — field-first approach
- MIT OCW, Electricity and Magnetism (8.02) — circuits and fields, https://ocw.mit.edu/
- John David Jackson, Classical Electrodynamics — advanced field reference (selective)
Every Wunder lesson is built from real, reputable sources — never invented.
Related courses
Wunder is a personalized learn-anything platform — tell it any topic and it builds a beautiful, fact-checked course in minutes, with narration, a knowledge check, and a college-style University track.
© 2026 Wunder Learning LLC · Terms & Privacy