Comprehensive theory, key formulas, diagrams, and memory aids for Magnetic Fields.
A magnetic field is a region of space in which a moving charged particle or a current-carrying conductor experiences a force. Magnetic fields are represented by field lines — the closer the lines, the stronger the field.
Magnetic flux density ($B$, unit: Tesla, T) quantifies the strength of a magnetic field. It is defined through the force on a current-carrying conductor: $$F = BIL\sin\theta$$
Where: - $F$ = force on conductor (N) - $B$ = magnetic flux density (T) - $I$ = current (A) - $L$ = length of conductor in the field (m) - $\theta$ = angle between the conductor and the magnetic field
Maximum force when conductor is perpendicular to field ($\theta = 90°$, $\sin\theta = 1$): $F = BIL$. Zero force when parallel to field ($\theta = 0°$).
The direction of the force is given by Fleming's Left Hand Rule (for conventional current): - Thumb → Force (thrust/motion) - Index finger → Field (north to south) - Middle finger → Current (conventional, + to -)
A moving charge in a magnetic field experiences a force: $$F = BQv\sin\theta$$
Where $Q$ = charge (C), $v$ = velocity (m s⁻¹). For $\theta = 90°$: $F = BQv$.
This force is always perpendicular to the velocity — it does no work on the charge and therefore does not change the particle's kinetic energy or speed. It only changes the direction, causing circular motion.
For a charged particle moving perpendicular to a uniform magnetic field: $$BQv = \frac{mv^2}{r} \quad \Rightarrow \quad r = \frac{mv}{BQ}$$
The radius of circular motion increases with momentum ($mv$) and decreases with field strength ($B$) and charge ($Q$).
Mass spectrometers use this principle: ions of different masses are deflected by different amounts in a magnetic field, allowing isotope separation and mass measurement.
graph TD
A[Charged Particle enters Magnetic Field] --> B[Force F = BQv perpendicular to v]
B --> C[Direction changes continuously]
C --> D[Circular motion: r = mv/BQ]
D --> E[Speed unchanged: F does no work]
Magnetic flux ($\Phi$) is a measure of the total magnetic field passing through a given area: $$\Phi = BA\cos\theta$$
Where $A$ = area (m²) and $\theta$ = angle between $B$ and the normal to the area. Unit: weber (Wb) = T m².
Faraday's Law of Electromagnetic Induction: An e.m.f. is induced in a conductor whenever there is a change in magnetic flux linkage ($N\Phi$, where $N$ = number of turns): $$\varepsilon = -\frac{d(N\Phi)}{dt} = -N\frac{d\Phi}{dt}$$
Lenz's Law: The direction of the induced e.m.f. (and resulting current) is such that it opposes the change in flux that caused it. This is a consequence of conservation of energy — you must do work against the opposing magnetic force to induce a current.
The negative sign in Faraday's Law represents Lenz's Law.
An e.m.f. is induced whenever there is relative motion between a conductor and a magnetic field, or when the magnetic flux through a coil changes. Practically:
Moving a conductor through a magnetic field: $\varepsilon = BLv$ (for a straight conductor of length $L$ moving at velocity $v$ perpendicular to field $B$).
Rotating a coil in a magnetic field: As the coil rotates, the angle between $B$ and the coil normal changes continuously, varying the flux linkage sinusoidally: $$N\Phi = NBA\cos(\omega t)$$ $$\varepsilon = NBA\omega\sin(\omega t) = \varepsilon_0\sin(\omega t)$$
This is the basis of the alternating current generator (AC generator). The peak e.m.f. is $\varepsilon_0 = NBA\omega$.
A transformer transfers electrical energy between two coils (primary and secondary) via a changing magnetic flux in a shared iron core: $$\frac{V_s}{V_p} = \frac{N_s}{N_p}$$
For an ideal (100% efficient) transformer: $$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$$
Also: $V_p I_p = V_s I_s$ (input power = output power, ideal case)
Why use high voltage for transmission? Power loss in transmission cables $P_{loss} = I^2 R$. By stepping up voltage (reducing current), power losses are dramatically reduced. The National Grid uses ~400 kV for long-distance transmission. Step-down transformers at substations reduce to safe domestic voltages (230 V in UK).
AC is produced by rotating generators. The voltage and current vary sinusoidally with time: $$V = V_0\sin(\omega t), \quad I = I_0\sin(\omega t)$$
For power calculations, we use root-mean-square (r.m.s.) values, which are the equivalent DC values that deliver the same average power: $$V_{rms} = \frac{V_0}{\sqrt{2}}, \quad I_{rms} = \frac{I_0}{\sqrt{2}}$$
Average power: $\bar{P} = V_{rms}I_{rms} = \frac{V_0 I_0}{2} = \frac{V_0^2}{2R}$
When current-carrying conductor (or semiconductor) is placed in a perpendicular magnetic field, charge carriers are deflected to one side, building up a charge. This creates the Hall voltage across the width of the material: $$V_H = \frac{BI}{ntq}$$
Where $n$ = charge carrier density, $t$ = thickness in direction of $B$, $q$ = carrier charge. The Hall effect is used in Hall probes to measure magnetic flux density and in sensors determining whether magnets are nearby (used in brushless motors and smartphones).