Magnetism

Benjamin Crowell · Simple Nature Ch. 7 · CC BY-SA 3.0

Moving charges create magnetic fields. Magnetic forces are always perpendicular to velocity, so they change direction without changing speed. Ampere's law relates the field around a wire to the current through it.

The magnetic field

Electric charges at rest create electric fields. Electric charges in motion also create magnetic fields. The magnetic field B is a vector field measured in teslas (T). Unlike electric field lines that start and end on charges, magnetic field lines always form closed loops.

Scheme

; Magnetic field of a long straight wire
; B = mu_0 * I / (2 * pi * r)
; mu_0 = 4*pi*1e-7 T*m/A (permeability of free space)

(define pi 3.14159265)
(define mu0 (* 4 pi 1e-7))

(define (B-wire I r)
  (/ (* mu0 I) (* 2 pi r)))

; 10 A current, field at various distances
(display "B at 1 cm: ")
(display (B-wire 10 0.01))
(display " T") (newline)

(display "B at 5 cm: ")
(display (B-wire 10 0.05))
(display " T") (newline)

(display "B at 10 cm: ")
(display (B-wire 10 0.10))
(display " T") (newline)

; Field falls off as 1/r
(display "Ratio B(1cm)/B(10cm): ")
(display (/ (B-wire 10 0.01) (B-wire 10 0.10)))

The Lorentz force

A charge q moving with velocity v through a magnetic field B feels a force F = qv x B. The cross product means the force is perpendicular to both the velocity and the field. A positive charge moving right in a field pointing into the page curves upward. This perpendicularity means magnetic forces do no work: they steer, but never speed up or slow down.

Scheme

; Lorentz force: F = q * v * B * sin(theta)
; For a charge moving perpendicular to B (theta = 90 deg):
; F = qvB, and the charge moves in a circle.
; Radius of circular motion: r = mv / (qB)

(define (lorentz-force q v B theta)
  (* q v B (sin theta)))

(define (cyclotron-radius m v q B)
  (/ (* m v) (* q B)))

; Proton in Earth's magnetic field (~50 microT)
(define q-proton 1.6e-19)    ; C
(define m-proton 1.67e-27)   ; kg
(define v-proton 1e6)        ; m/s (1000 km/s)
(define B-earth 50e-6)       ; T

(define F (lorentz-force q-proton v-proton B-earth (/ 3.14159 2)))
(display "Force on proton: ")
(display F) (display " N") (newline)

(define r (cyclotron-radius m-proton v-proton q-proton B-earth))
(display "Cyclotron radius: ")
(display r) (display " m") (newline)
(display "That's about ")
(display (/ r 1000))
(display " km")

Ampere's law

Ampere's law says the line integral of B around any closed loop equals mu_0 times the current passing through the loop. In symbols: the integral of B dot dl around a closed path = mu_0 * I_enclosed. This is the magnetic analog of Gauss's law. It's most useful when symmetry lets you pull B out of the integral.

Scheme

; Ampere's law: integral of B . dl = mu0 * I_enclosed
; For a circular Amperian loop around a wire:
;   B * 2*pi*r = mu0 * I
;   B = mu0*I / (2*pi*r)  -- same result as before!

(define pi 3.14159265)
(define mu0 (* 4 pi 1e-7))

; Verify: line integral of B around a circle of radius r
(define (ampere-check I r)
  (let ((B (/ (* mu0 I) (* 2 pi r))))
    (* B 2 pi r)))

; Should equal mu0 * I regardless of r
(define I 5.0)
(display "mu0 * I = ") (display (* mu0 I)) (newline)
(display "Loop integral (r=0.01): ") (display (ampere-check I 0.01)) (newline)
(display "Loop integral (r=0.10): ") (display (ampere-check I 0.10)) (newline)
(display "Loop integral (r=1.00): ") (display (ampere-check I 1.00)) (newline)
(display "All equal: Ampere's law confirmed.")

Solenoids

A solenoid is a coil of wire. Inside a long solenoid the field is uniform and parallel to the axis: B = mu_0 * n * I, where n is the number of turns per meter. Outside, the field is nearly zero. Solenoids are how we make controllable, uniform magnetic fields.

Scheme

; Solenoid: B = mu0 * n * I
; n = N/L (turns per meter)

(define pi 3.14159265)
(define mu0 (* 4 pi 1e-7))

(define (B-solenoid n I)
  (* mu0 n I))

; 500 turns over 0.2 m, carrying 3 A
(define N 500)
(define L 0.2)
(define n (/ N L))
(define I 3.0)

(display "Turns per meter: ") (display n) (newline)
(display "B inside solenoid: ")
(display (B-solenoid n I))
(display " T") (newline)

; Compare to Earth's field
(display "Times stronger than Earth's field: ")
(display (/ (B-solenoid n I) 50e-6))

Magnetic flux

Magnetic flux through a surface is the integral of B dot dA. For a uniform field through a flat loop: Phi = B * A * cos(theta). Flux is measured in webers (Wb = T * m^2). Gauss's law for magnetism says the total flux through any closed surface is zero. There are no magnetic monopoles: every field line that enters a surface also exits.

Scheme

; Magnetic flux: Phi = B * A * cos(theta)
; Gauss's law for magnetism: total flux through closed surface = 0

(define pi 3.14159265)

(define (magnetic-flux B A theta)
  (* B A (cos theta)))

; Circular loop of radius 5 cm in a 0.1 T field
(define r 0.05)
(define A (* pi r r))
(define B 0.1)

; Flux at various angles
(display "Flux at 0 deg (face-on): ")
(display (magnetic-flux B A 0))
(display " Wb") (newline)

(display "Flux at 45 deg: ")
(display (magnetic-flux B A (/ pi 4)))
(display " Wb") (newline)

(display "Flux at 90 deg (edge-on): ")
(display (magnetic-flux B A (/ pi 2)))
(display " Wb") (newline)

; At 90 degrees, no field lines pass through
(display "Edge-on flux is zero: ")
(display (< (abs (magnetic-flux B A (/ pi 2))) 1e-15))

Neighbors

Cross-references

Calculus Ch.14 — line integrals: the math behind Ampere's law

Foundations (Wikipedia)

Translation notes

All formulas assume SI units. The cross product in the Lorentz force is computed as a scalar magnitude here (q*v*B*sin theta) since our Scheme REPL lacks vector operations. Crowell develops the vector form in detail. The solenoid formula assumes an ideal infinite solenoid; real solenoids have fringe fields at the ends.

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