Electromagnetic Induction - Quick Revision
Magnetic flux
- Definition: phi_B = B.A = B A cos(theta), where theta is the angle between B and the area vector (normal to the surface).
- SI unit: weber (Wb) = T m^2. Magnetic flux is a scalar.
- Flux can be changed by varying B, the area A, or the angle theta (e.g. rotating or deforming a coil).
Faraday's law
- An emf is induced whenever the flux through a circuit changes with time: emf = -dphi_B/dt.
- For a coil of N turns: emf = -N dphi_B/dt; more turns give a larger emf.
- In a closed circuit of resistance R the induced current is I = emf/R.
- A steady (unchanging) flux induces no emf, however large the field.
Lenz's law and energy
- The induced emf opposes the change in flux that produces it (this is the negative sign).
- It is a consequence of conservation of energy: work done against the opposing force appears as Joule heating; a perpetual-motion machine is impossible.
Motional emf
- A rod of length l moving at speed v perpendicular to B has emf = B l v (Lorentz-force view: charges feel qvB).
- A rod rotating about one end: emf = (1/2) B omega L^2.
Inductance
- Self-inductance: N phi = L I; back emf = -L dI/dt; it opposes change in the coil's own current (electrical inertia, analogue of mass).
- Energy stored: W = (1/2) L I^2; energy density u = B^2/(2 mu0).
- Mutual inductance: N1 phi1 = M I2; emf_1 = -M dI_2/dt; M12 = M21.
- Long solenoid: L = mu_r mu0 n^2 A l (n = turns per metre).
AC generator
- A coil of N turns, area A rotating at angular speed omega in field B gives emf = NBA omega sin(omega t); peak emf e0 = NBA omega.
- Converts mechanical energy to electrical energy; frequency 50 Hz in India.