Everything in this course rests on four laws. Each began as an experimental discovery: Coulomb's measurements of the electric force, the observation that magnets always have two poles, Faraday's induction, and Ampère's work on currents, completed by Maxwell's displacement current. Together with the Lorentz force, they describe all of classical electricity, magnetism and light.
| Law | Equation | What it says |
|---|---|---|
| Gauss's law | $\oint\vec{E}\cdot d\vec{A} = \dfrac{q_{\text{enc}}}{\varepsilon_0}$ | Charges are sources of electric field; flux out of a closed surface counts the charge inside. |
| Gauss's law for magnetism | $\oint\vec{B}\cdot d\vec{A} = 0$ | There are no magnetic charges: field lines never start or end. |
| Faraday's law | $\oint\vec{E}\cdot d\vec{\ell} = -\dfrac{d\Phi_B}{dt}$ | A changing magnetic field makes a curling electric field. |
| Ampère–Maxwell law | $\oint\vec{B}\cdot d\vec{\ell} = \mu_0 I_{\text{enc}} + \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$ | Currents and changing electric fields make curling magnetic fields. |
Key idea
Two laws are about flux through closed surfaces (the sources of $\vec{E}$ and $\vec{B}$), and two are about circulation around loops (what makes the fields curl). Changing $\vec{B}$ makes $\vec{E}$; changing $\vec{E}$ makes $\vec{B}$. With $\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$ for how fields push charges, that is the whole theory.
Why it works
Notice the near-symmetry. If magnetic charges existed, the second law would have a charge term like the first, and Faraday's law would gain a "magnetic current" like Ampère's. None has ever been found, despite careful searches, so the magnetic flux through every closed surface is zero: every field line that enters must leave. The two curl laws mirror each other with a sign difference, and that sign is what makes waves possible instead of runaway growth.
Worked examples
Worked example: Closed-surface bookkeeping
The outward magnetic flux through the flat top of a closed cylinder is $5.0\ \text{mWb}$, and through its flat bottom $-2.0\ \text{mWb}$. What is the flux through its curved side?
- $\oint\vec{B}\cdot d\vec{A} = 0$, so the three parts add to zero.
- $5.0 + (-2.0) + \Phi_{\text{side}} = 0$, so $\Phi_{\text{side}} = -3.0\ \text{mWb}$ (more field enters through the side than leaves).
Answer: $-3.0\ \text{mWb}$
Worked example: Which law?
Name the law behind each: (a) a transformer, (b) the field around the gap of a charging capacitor, (c) a lone north pole cannot exist, (d) the field outside a charged sphere is $kQ/r^2$.
- (a) Faraday's law: the changing flux of the primary induces an emf in the secondary.
- (b) The Ampère–Maxwell law, through its displacement-current term.
- (c) Gauss's law for magnetism. (d) Gauss's law.
Answer: Faraday; Ampère–Maxwell; Gauss for magnetism; Gauss
Worked example: Both kinds of current
A path encloses a wire carrying $2.0\ \text{A}$ and part of a capacitor gap where the electric flux grows at $1.0 \times 10^{11}\ \text{V·m/s}$. Find $\oint\vec{B}\cdot d\vec{\ell}$.
- Displacement current: $\varepsilon_0\dfrac{d\Phi_E}{dt} = (8.85 \times 10^{-12})(1.0 \times 10^{11}) = 0.885\ \text{A}$.
- $\oint\vec{B}\cdot d\vec{\ell} = \mu_0(2.0 + 0.885) = 3.63 \times 10^{-6}\ \text{T·m}$.
Answer: $3.63 \times 10^{-6}\ \text{T·m}$
Common mistake
"The magnetic flux through a closed surface can be nonzero, if the surface surrounds a magnet's north pole." That would require a lone magnetic charge. A real magnet's field lines run back through the magnet, so a surface around one end catches them coming out and going in. Check: any closed surface, anywhere, has zero net magnetic flux.