Electromagnetism G. L. Pollack and D. R. Stump 10. Electromagnetic Induction Chapter Summary Faraday’s law If a magnetic field changes in time, there is an induced electric field. The field equation that describes the phenomenon of electromagnetic induction is ∇× E = − ∂B , ∂t which is called Faraday’s law. The integral form is I dΦ E= E · d` = − dt C where Φ is the magnetic flux through any surface with boundary curve C. Lenz’s law The direction of E in electromagnetic induction opposes the change of flux; i.e., if a conductor is present then the induced current produces a magnetic field in the direction that tends to maintain the flux. Self-inductance. A current I in a conducting loop creates a magnetic field. The flux through the loop is proportional to the current, Φ = LI. The constant of proportionality L is the self-inductance, which depends on the geometry of the conducting loop. If I changes in time, there is an induced emf around the loop, and by Faraday’s law, E = −L Mutual inductance dI . dt For 2 loops, Φ12 = M I2 and Φ21 = M I1 . By Faraday’s law, the emfs induced around the loops are E1 = −L1 I˙1 − M I˙2 E2 = −L2 I˙2 − M I˙1 The energy density of the magnetic field is umag = B2 . 2µ0 Motional emf If a conducting loop moves in a static magnetic field, the emf around the loop is E = −dΦ/dt.