Two laws, two new verbs
Rung 1 left you holding two field maps: arrows that show which way and how hard a tiny test charge would be shoved (the electric field E), and arrows that show which way a compass needle would swing (the magnetic field B). Beautiful pictures — but a picture is not a law. A law has to answer the next question a curious 12-year-old always asks: *but where do the arrows come from?* That is exactly what Gauss and Ampère answer.
Both laws share a sneaky structure. Each one wraps an imaginary surface or loop around something, then says: *the field, summed up over that boundary, equals the stuff trapped inside.* The field is the symptom; the trapped stuff is the cause. So before the physics, learn two verbs — because once you have the verbs, the laws read like plain English.
- Flux — *how much field pierces a surface.* Imagine holding a tennis racket in a rainstorm. Flux is the rate water passes through the strings. Hold it flat to the rain: lots of flux. Turn it edge-on: almost none, even in the same downpour. Flux cares about strength and angle.
- Circulation — *how much field loops around a closed path.* Imagine walking a full lap around a pond and at every step noting how much the wind pushes you along your direction of travel, then adding it all up. If the air is swirling, you come back with a nonzero total. That running total around a loop is circulation.
Gauss's law: field lines burst out of charge
Picture a single positive charge floating in space, sprouting electric field lines in every direction like the spikes of a sea urchin. Now blow up a balloon around it — any shape, any size, as long as it fully encloses the charge. Gauss's law makes a claim that sounds almost magical: *count the field lines piercing the balloon's skin, and that count depends only on the charge inside — not on the balloon's size, not on its shape, not on where the charge sits inside it.*
Why does shape not matter? Because field lines don't start or stop in empty space — they only begin on positive charge and end on negative charge. So every line that leaves the charge *must* cross the balloon's surface exactly once on its way out. Dent the balloon inward and a line might pierce it three times — out, in, out — but the ins and outs cancel and the net is still one. The total flux is just bookkeeping on an unbreakable rule: lines have nowhere to hide.
Gauss's law (the words):
( total E-field flux out of any closed surface ) = ( charge enclosed ) / epsilon_0
___________
/ \ +q <- the only thing that matters
| /|\ | is what's INSIDE the bag
| / | \ ----+--> E epsilon_0 = 8.854e-12 F/m
| +q lines | (permittivity of free space)
| \ | / |
\___\|/______/
closed surface
(any shape!)Here is where the magic pays off. Because the flux is the same through any enclosing surface, you get to *choose* the most convenient one — and for a point charge the obvious choice is a sphere centered on it, because by symmetry E has the same strength everywhere on that sphere and points straight out. The total flux is then just (field strength) × (area of sphere). Setting that equal to the enclosed-charge rule and solving for the field gives the famous inverse-square law for free:
Worked example — field of a point charge q flux = E * (area of sphere) = E * 4*pi*r^2 Gauss: E * 4*pi*r^2 = q / epsilon_0 => E = q / (4*pi*epsilon_0*r^2) (points radially outward) Double the distance r -> quarter the field. That's 1/r^2. Numbers: q = 1 nC = 1e-9 C, at r = 0.10 m E = (9e9) * (1e-9) / (0.10)^2 ~= 900 V/m [ k = 1/(4*pi*eps0) ~= 9e9 ]
Inside vs. outside a charged sphere
Gauss's law really earns its keep when the charge is spread out. Take a solid ball of radius R with charge sprinkled uniformly through its volume — a decent stand-in for a charged metal bead or a model nucleus. We want the field at every distance r. The trick never changes: draw an imaginary sphere ('a Gaussian surface') of radius r, and ask only one question — *how much charge does it enclose?*
- Outside (r > R): your Gaussian sphere encloses *all* the charge Q. The math is identical to the point charge — the ball might as well be a single point at its center. So E = Q / (4πε₀r²). From the outside, you cannot tell a uniformly charged ball from a point charge. That is a deep fact, not a coincidence.
- Inside (r < R): your Gaussian sphere only catches the charge *within radius r* — a fraction (r/R)³ of the total, because charge scales with the volume you've enclosed. The shell of charge outside r contributes exactly zero field inside (its pulls cancel perfectly by symmetry). Plug that smaller enclosed charge into Gauss and you get E = Q·r / (4πε₀R³): the field grows linearly from zero at the center.
Field of a uniformly charged ball, radius R, total charge Q E | . | .' '. outside: E ~ 1/r^2 (falls off) | .' ' . | .' ' . _ | .' ' - . _ | .' inside: E ~ r ' - . _ _ _ _ | .' (rises straight) | .' +------------------|------------------------------> r 0 R Peak field sits right at the surface, r = R.
Ampère's law: field lines wrap around current
Now flip the script from charge to motion. Charge sitting still makes an electric field; charge *moving* — that is, current — makes a magnetic field. And the magnetic field does something the electric field never does: instead of bursting outward from a source, it *circles* the source. Wrap your right hand around a wire with your thumb pointing along the current, and your curled fingers point the way B swirls. The field has no beginning and no end — it just loops.
Ampère's law is the circulation bookkeeping for exactly this. Draw an imaginary loop in space (an 'Amperian loop'), walk all the way around it adding up how much B pushes you along your path, and the law says that running total equals the *current threaded through* the loop. It is the perfect mirror of Gauss: Gauss counts charge poked *through a surface*; Ampère counts current *threaded through a loop*.
Ampere's law (the words):
( B-field circulation around any closed loop ) = mu_0 * ( current threaded )
current I out of page
.
( o ) B circles around it
/ | \ (right-hand rule: thumb = I, fingers = B)
| | |
\ | / <-- Amperian loop: walk it, sum B along the path
'-+-'
mu_0 = 4*pi*1e-7 T*m/A (permeability of free space)Worked example — the long straight wire. By symmetry, B must have the same strength all the way around any circle centered on the wire, and it points along that circle. So choose your Amperian loop to *be* that circle of radius r. The circulation is just (field strength) × (circumference), and the current threaded is simply I. Solve:
Long straight wire carrying current I
circulation = B * (circumference) = B * 2*pi*r
Ampere: B * 2*pi*r = mu_0 * I
=> B = mu_0 * I / (2*pi*r) (circles the wire, 1/r fall-off)
Numbers: I = 10 A, at r = 0.05 m (5 cm away)
B = (4*pi*1e-7 * 10) / (2*pi*0.05)
= (2e-7 * 10) / 0.05 = 4e-5 T = 40 microtesla
(~ same order as Earth's field, ~50 uT — measurable with a phone compass!)The solenoid: bottling a uniform field
The single wire is cute, but engineers want a *strong, uniform, controllable* magnetic field — for motors, MRI scanners, relays, the deflection coils in old TVs. The answer is the solenoid: take that wire and wind it into a tight helix, hundreds of turns long. Each turn's field adds to its neighbours' inside the coil and largely cancels outside. The result is a field that is nearly uniform down the bore and nearly zero outside — a bottle of magnetic field, with a tap you control via the current.
Ampère's law makes this rigorous with a clever rectangular loop: run one long side straight down the inside of the solenoid (where B is strong and uniform), bring the other long side back outside (where B ≈ 0), and the two short ends are perpendicular to the field so they contribute nothing. Only the inside leg adds up. Lasso the current and you find the field depends only on how *tightly* you wind, not on the coil's diameter:
Ideal solenoid: n turns per metre, current I B (inside) = mu_0 * n * I B (outside) ~= 0 |---- only this inside leg has B ----| +====================================+ || -> -> -> -> -> -> -> -> || B uniform, points along axis +====================================+ |<-- Amperian rectangle, outside leg sees B~=0 -->| Numbers: 1000 turns over 0.5 m -> n = 2000 turns/m, I = 2 A B = (4*pi*1e-7) * 2000 * 2 ~= 5.0e-3 T = 5 mT (~100x Earth's field, from a couple of amps and a spool of wire)
The asymmetry — and what's still missing
Step back and look at the two laws side by side and you'll feel a strange imbalance. Gauss says electric field lines *start and stop* — they erupt from positive charge and plunge into negative charge. They have ends. Ampère says magnetic field lines *never start or stop* — they only form closed loops around current. They have no ends. Why the difference?
Because nature gives you isolated electric charges — a lone proton, a lone electron — but, as far as anyone has ever found, *no isolated magnetic charge*. Snap a bar magnet in half and you don't get a north pole and a south pole in separate hands; you get two smaller magnets, each with its own north and south. There is no 'magnetic monopole' for field lines to start on, so they have no choice but to close on themselves. That single fact is the deepest asymmetry in classical electromagnetism.
Here is the cliff-hanger. Rung 3 brings Faraday's law: a *changing* magnetic field stirs up an electric field — the principle behind every generator and transformer. Then Maxwell patches a subtle leak in Ampère's law (try applying it to a charging capacitor and the bookkeeping breaks!) by adding a term where a *changing* electric field makes a magnetic field. Suddenly E and B can create each other with no charge or current in sight — and that self-sustaining handshake, racing through empty space, *is* light. Two static source laws today; a unified electrodynamics — the four Maxwell equations — by the end of this track.