Fields

Fields describe the region of space around an object where another object experiences a force without physical contact. This section covers gravitational fields, electric fields, and magnetic fields — and their unification through electromagnetic induction.

Topics Covered

Gravitational Fields

• Newton’s law of gravitation — $F = \frac{GMm}{r^2}$; inverse square law
• Gravitational field strength — $g = \frac{GM}{r^2}$; radial fields around point masses; uniform fields near Earth’s surface ($g \approx 9.81\,\text{N/kg}$)
• Gravitational potential — $V_g = -\frac{GM}{r}$; potential energy $E_p = -\frac{GMm}{r}$; work done moving mass in a field
• Orbits — circular orbits: $F = \frac{mv^2}{r}$; geostationary and polar orbits; escape velocity $v_{\text{esc}} = \sqrt{\frac{2GM}{r}}$

Electric Fields

• Coulomb’s law — $F = \frac{Q_1 Q_2}{4\pi\varepsilon_0 r^2}$; similarity to gravitational force but with charge
• Electric field strength — $E = \frac{F}{Q} = \frac{Q}{4\pi\varepsilon_0 r^2}$ (radial); $E = \frac{V}{d}$ (uniform between plates)
• Electric potential — $V_e = \frac{Q}{4\pi\varepsilon_0 r}$; potential energy $E_p = \frac{Q_1 Q_2}{4\pi\varepsilon_0 r}$
• Uniform fields — between parallel plates; force on charge $F = QE$; motion of charged particles (parabolic paths analogous to projectile motion)
• Comparison: gravitational vs. electric fields — both follow inverse square laws, but gravity is always attractive while electric fields can be attractive or repulsive

Magnetic Fields

• Magnetic flux density — $B$; the tesla; $F = BIl\sin\theta$ for current-carrying conductors
• Fleming’s left-hand rule — determining force direction on a current in a magnetic field
• Charged particles in magnetic fields — circular motion with radius $r = \frac{mv}{BQ}$; frequency independent of speed
• Magnetic flux and flux linkage — $\Phi = BA\cos\theta$; $N\Phi = BAN\cos\theta$

Electromagnetic Induction

• Faraday’s law — $\varepsilon = -\frac{d(N\Phi)}{dt}$; induced EMF equals rate of change of flux linkage
• Lenz’s law — the induced current opposes the change producing it; conservation of energy
• AC generator — $\varepsilon = BAN\omega\sin(\omega t)$; peak EMF and RMS values
• Transformers — $\frac{V_s}{V_p} = \frac{N_s}{N_p}$; efficiency and power transmission

Study Tips

1. Compare gravitational and electric fields explicitly — learn the parallels (inverse square laws, potential equations) and the differences (attractive only vs. attractive/repulsive, mass vs. charge).
2. Use Fleming’s left-hand rule physically — actually hold your left hand in the correct orientation. Practise until it’s automatic.
3. Derive orbital velocity from combining $\frac{GMm}{r^2}$ with $\frac{mv^2}{r}$ — this derivation appears frequently.
4. Practise Lenz’s law — for any situation, ask “what change is happening?” and then “what current would oppose this change?”
5. Sketch field lines — radial (point mass/charge), uniform (between plates), and the combined fields.

How to Use These Notes

Follow the sidebar order. Each page provides physical principles, derivations from first principles, worked examples, and exam-style problems. Start with gravitational fields, then electric fields, then magnetic fields and electromagnetic induction.