Electric & Magnetic Fields · FieldLab

Field Lines & Equipotentials

The field of a point charge is E = kQ/r²; the fields of several charges superpose as vectors. Electric field lines picture the field: the tangent at each point gives the field direction, and the line density shows the field strength. Field lines start on positive charges (or at infinity) and end on negative charges (or at infinity); they never cross and never close on themselves.

The electric potential φ = kQ/r is a scalar and simply adds up. Points of equal potential form equipotential lines:

Equipotentials are everywhere perpendicular to field lines; potential decreases along a field line

Drag the charges to move them, and drag the probe to read E and φ at any point. Turn on the 3D potential surface: positive charges are "peaks", negative charges are "valleys", and field lines follow the "steepest way downhill".

Work by the Field & Potential Energy

When a test charge q moves from A to B, the work done by the electric field is

WAB = q (φA − φB) = q UAB

It depends only on the start and end points, not on the path — just like the work done by gravity. So we can define the electric potential energy Ep = qφ: when the field does positive work, the potential energy decreases; when it does negative work, the potential energy increases.

Move q along three paths — straight, zigzag and curved. The W - s curves in the data dock differ, yet they all meet at the same point at the end. Drag points A and B, or switch the test charge to a negative charge and try again.

Parallel-Plate Capacitor

Two parallel metal plates, close together and insulated from each other, form a parallel-plate capacitor. Its capacitance is

C = εr S / (4πkd) = εrε0 S / d

After charging, disconnect the source: the charge Q on the plates stays constant, and from U = Q/C, increasing d, reducing the overlapping area S, or removing the dielectric makes C smaller and the deflection of the electrometer needle (U) larger. The field between the plates, E = U/d = Q/(εrε0S), does not depend on d.

If the capacitor stays connected to the source, U is constant, and Q and E change as d changes. Record several data sets: the C - 1/d graph is a straight line through the origin.

Cathode-Ray Tube

Heated by the filament, cathode K emits electrons, which are accelerated by the accelerating voltage U₁: eU₁ = ½mv₀². The electrons enter the deflection plates (length L, separation d) at speed v₀, perpendicular to the field, and follow a projectile-like motion in the field. On leaving the plates, the sideways displacement is

y = qU₂L² / (2mdv₀²) = U₂L² / (4dU₁)

They then travel in a straight line at constant speed to the fluorescent screen, as if shot straight from the midpoint of the plates. The deflection is proportional to U₂ — this is how an oscilloscope measures voltage.

A sawtooth sweep voltage on the X plates sweeps the beam across the screen from left to right at constant speed; with the signal on the Y plates, the screen draws the signal's waveform.

Magnetic Field Lines

Sprinkle an even layer of iron filings on a glass plate and tap it gently: the magnetized filings line up along the field into curves. Place a compass needle: when it comes to rest, its N pole points in the direction of the field at that point.

Magnetic field lines run from the N pole to the S pole outside a magnet and from S to N inside it — they are closed curves.

Electric currents also produce magnetic fields (Oersted). The field lines of a straight wire are concentric circles; their direction is given by Ampère's rule (the right-hand grip rule):

Straight wire B = μ₀I / (2πr)

The fields of a current loop and of a current-carrying solenoid resemble that of a bar magnet; inside the solenoid the field is nearly uniform.

Ampère Force

A copper rod lies across two horizontal metal rails between the poles of a horseshoe magnet. Close the circuit: current flows through the rod, and the Ampère force makes it roll along the rails.

F = B I L

L is the effective length of the conductor in the field (more magnets give a longer L). The direction of the Ampère force follows the left-hand rule: hold your left hand flat so that the field lines enter the palm, with the four fingers pointing along the current; the thumb then points in the direction of the force.

Reversing either the current or the magnetic poles reverses the rod's motion; reversing both leaves the direction unchanged.

Lorentz Force

A charge moving in a magnetic field feels the Lorentz force F = qvB (v ⊥ B); its direction follows the left-hand rule (opposite for negative charges). The Lorentz force is always perpendicular to the velocity, so it does no work and only changes the direction of the velocity.

Electron beam tube: Helmholtz coils produce a uniform field, and the thin gas in the glass bulb makes the beam's path glow. An electron entering perpendicular to the field moves in uniform circular motion:

r = mv / (qB) T = 2πm / (qB)

Tilt the electron gun so the velocity has a component parallel to B, and the path becomes a helix.

Velocity selector: with perpendicular electric and magnetic fields, only particles with qE = qvB, i.e. v = E / B, pass straight through the slit.

Mass Spectrometer & Cyclotron

Mass spectrometer: ions accelerated through voltage U enter a uniform magnetic field perpendicularly through slit S, travel half a circle and strike the photographic plate:

qU = ½mv² r = mv/(qB) = √(2mU/q) / B

For isotopes with the same charge, the heavier the ion, the larger the radius, so they form separate spectral lines on the plate, with r₁ : r₂ = √m₁ : √m₂. This is how Aston discovered the isotopes of neon.

Cyclotron: two D-shaped electrodes (dees) sit in a uniform magnetic field with an alternating voltage across the gap. The particle is accelerated each time it crosses the gap and its radius keeps growing, but the period T = 2πm/(qB) is independent of speed, so the frequency of the alternating voltage can stay fixed. The final kinetic energy

Ek = q²B²R² / (2m)

depends only on B and the dee radius R, not on the accelerating voltage.

Electromagnetic Induction & Lenz's Law

When the magnetic flux through a closed circuit changes, an induced current flows in it — electromagnetic induction (Faraday). The induced emf is

E = N ΔΦ / Δt

Drag the magnet into and out of the coil: the galvanometer needle deflects; when the magnet stops, the needle returns to zero. The faster you move it, the larger the deflection.

Lenz's law: the magnetic field of the induced current always opposes the change in flux that produces it — "repel on approach, attract on withdrawal".

A conducting rod cutting magnetic field lines: E = BLv. The induced current produces an Ampère force on the rod opposite to its motion, F = B²L²v/R, so under a constant pulling force the rod eventually moves at constant speed.

AC Generator

A rectangular coil rotates at constant speed in a uniform field about an axis perpendicular to the field. Sides ab and cd cut the field lines, producing an alternating current whose magnitude and direction vary periodically. Timing from the neutral plane:

e = Em sin ωt Em = NBSω

The neutral plane is where the coil plane is perpendicular to the field: here the flux is at its maximum, but no side cuts the field lines, so the emf is zero. After 90° the flux is zero and the emf is maximum. Each time the coil passes the neutral plane, the current reverses direction.

The RMS (effective) value of a sinusoidal AC is E = Em / √2: an AC and a DC passing through identical resistors for the same time produce the same heat.

Sandbox

Place point charges freely on the bench, turn on a uniform magnetic field region on the right half, and fire electrons, protons or α particles from the particle gun to see their paths in electric and magnetic fields (computed step by step with the Boris algorithm).

· Click the buttons on the panel to add charges; drag a charge to move it; select it and press Delete to remove it· Drag the particle gun to move it; drag the gold ring to change the firing direction· Particle motion plays in slow motion

Try making an electron orbit a positive charge, or making a proton turn half a circle in the field and come back.