Master Formula Sheet

Everything computable in this chapter, on one card:

# Result Formula
1 Magnetic flux ΦB=B⃗⋅A⃗=BAcos⁡θ\Phi_B = \vec{B}\cdot\vec{A} = BA\cos\theta (weber = T m2^2 = V s)
2 Faraday's law ε=−NdΦBdt\varepsilon = -N\dfrac{d\Phi_B}{dt}
3 Induced charge q=NΔΦBRq = \dfrac{N\Delta\Phi_B}{R} (rate-independent)
4 Motional emf (sliding rod) ε=Blv\varepsilon = Blv
5 Rotating rod / spoke ε=12BωR2=πBνR2\varepsilon = \frac{1}{2}B\omega R^2 = \pi B\nu R^2
6 Force on rails rod F=BIl=B2l2vRF = BIl = \dfrac{B^2l^2v}{R}; power Fv=I2RFv = I^2R
7 Mutual inductance (def.) N1Φ1=MI2N_1\Phi_1 = MI_2; ε1=−M dI2dt\varepsilon_1 = -M\,\dfrac{dI_2}{dt}; M12=M21M_{12} = M_{21}
8 Coaxial solenoids M=μ0n1n2πr12lM = \mu_0 n_1 n_2 \pi r_1^2 l (inner radius!)
9 Concentric coils M=μ0πr12/2r2M = \mu_0\pi r_1^2/2r_2
10 Self-inductance (def.) NΦB=LIN\Phi_B = LI; back emf ε=−L dIdt\varepsilon = -L\,\dfrac{dI}{dt}
11 Long solenoid L=μ0n2AlL = \mu_0 n^2 Al (× μr\times\,\mu_r with core)
12 Stored energy W=12LI2W = \frac{1}{2}LI^2; density uB=B2/2μ0u_B = B^2/2\mu_0
13 AC generator ε=NBAωsin⁡ωt\varepsilon = NBA\omega\sin\omega t; ε0=NBAω\varepsilon_0 = NBA\omega

Units: weber (flux), henry (inductance) = Wb/A = V s/A.

The Direction Toolkit (Lenz in Three Steps)

  1. Find the field direction through the loop and decide: is ΦB\Phi_B increasing or decreasing?
  2. The induced current creates flux opposing that change (supports a dying flux, fights a growing one).
  3. Fix the current's sense with the right-hand rule.

Instant verdicts worth memorising:

  • Magnet N-pole approaching a coil: near face turns N (repels); receding: near face turns S (attracts).
  • Loop entering an into-page field: current anticlockwise; leaving: clockwise; fully inside: zero.
  • Magnet through a closed ring: falls with a<ga < g; through a cut ring: a=ga = g (emf yes, current no).
  • Lenz's law = conservation of energy: the work done against the opposition becomes Joule heat.

And the deepest line: for a stationary conductor, induction means a time-varying magnetic field generates an electric field — electricity and magnetism are one subject.

The Inertia Dictionary & Quick Comparisons

L as electrical mass:

Mechanics Electromagnetism
mass mm self-inductance LL
velocity vv current II
force F=m dv/dtF = m\,dv/dt back emf ∣ε∣=L dI/dt|\varepsilon| = L\,dI/dt
kinetic energy 12mv2\frac{1}{2}mv^2 magnetic energy 12LI2\frac{1}{2}LI^2

Field energy twins: uB=B22μ0u_B = \dfrac{B^2}{2\mu_0} (solenoid-derived, fully general) ↔\leftrightarrow uE=12ε0E2u_E = \frac{1}{2}\varepsilon_0 E^2 (capacitor-derived, fully general).

Inductance is like capacitance: both are constants set purely by geometry and the medium (μr\mu_r multiplies MM and LL; the dielectric constant plays the same role for CC), never by the current or charge they hold.

Generator phase facts: flux ∝cos⁡ωt\propto \cos\omega t, emf ∝sin⁡ωt\propto \sin\omega t — emf peaks when flux is zero (coil plane parallel to BB) and vanishes when flux peaks; average emf over a full cycle is zero, over a half cycle 2ε0/π2\varepsilon_0/\pi.

One-Glance Revision Flow

The story of the chapter in six steps:

  1. Faraday & Henry's experiments: current is induced only while the flux through a coil is changing — by relative motion or by switching a neighbour's current (no motion needed!).
  2. Faraday's law: ε=−N dΦB/dt\varepsilon = -N\,d\Phi_B/dt, with three handles on ΦB=BAcos⁡θ\Phi_B = BA\cos\theta: change B, change A, or change θ\theta.
  3. Lenz's law: the induced current opposes the change — energy conservation; direction in three steps.
  4. Motional emf: BlvBlv for a sliding rod (Lorentz force on the free charges), 12BωR2\frac{1}{2}B\omega R^2 for a rotating one; power in = heat out.
  5. Inductance: neighbours couple via MM (ε1=−M dI2/dt\varepsilon_1 = -M\,dI_2/dt, reciprocity M12=M21M_{12} = M_{21}); a coil resists itself via LL (back emf, 12LI2\frac{1}{2}LI^2 stored, B2/2μ0B^2/2\mu_0 per unit volume).
  6. AC generator: rotate the coil — ε=NBAωsin⁡ωt\varepsilon = NBA\omega\sin\omega t — and civilisation switches on.

Morning-of-exam checklist: flux zero when plane parallel to B … charge NΔΦ/RN\Delta\Phi/R is rate-independent … rotating rod has the 12\frac{1}{2}, sliding rod doesn't … spokes in parallel change nothing … M and L never depend on current … L∝n2L \propto n^2 … emf peaks where flux vanishes … cut ring: emf yes, current no, opposition no.

Now go score — this chapter converts revision into marks as efficiently as a generator converts motion into light.