Electromagnetism: Inductance, LC Circuits, and Waves, Study notes of Physics

Download Electromagnetism: Inductance, LC Circuits, and Waves and more Study notes Physics in PDF only on Docsity! Problems 34.5, 34.7, 34.15, 34.17, 34.31, 32.5, 32.7, 32.11, 32.53, 32.47, 32.51 Students wishing to not take the final exam must turn in the underlined problems to me before Wednessday. If the quality of the homework is not sufficiently high they will not be exempt. Inductance • For a typical coil, the current in changing, the resulting B field is changing, the changing B-field produces a changing magnetic flux, and a changing magnetic flux produces a voltage known as the back emf E = −L dI dt (1) The inducance in any coil is L = NΦB I (2) where ΦB is the magnetic flux through the coil for a fixed current and N is the number of turns. • For an ideal solenoid the inductance is L = µoN 2A ` (3) Here N is the the number of turns, A is the cross sectional area, ` is the length of solenoid. You should be able to derive this result. • The Energy stored in an inductor is U = 1 2 LI2 (4) The energy is stored within the magnetic field. The energy per unit volume is uB = B2 2µo (5) • LC circuits – In an L,C circuit that has zero resistance the current and charge on the capacitor change as Q = Qmax cos(ωot + φ) (6) I = dQ dt = −ωoQmax︸ ︷︷ ︸ Imax sin(ωot + φ) (7) where Qmax is the maximum charge on the capacitor and ωo = 1√ LC (8) is the oscillation frequency of the circuit – The energy in the LC circuit is constant and is the energy stored in the U = 1 2 LI2(t) + 1 2 Q2(t) C (9) Note at certain moments there is no current and only charge. At other moments there is only current and no charge, so U = 1 2 LI2max = 1 2 Q2max C (10) 1 Waves • In free space the electric field and magnetic field obey the wave equation. ∂2E ∂x2 = µoo ∂E ∂t2 (11) ∂2B ∂x2 = µoo ∂B ∂t2 (12) • The solution to this equation is E = Emax cos(kx− ωt) (13) B = Bmax cos(kx− ωt) (14) with k = 2π λ ω = 2πf c = λf (15) • The magnitude of the electric field and the magnitude of the magnetic field are related E = cB (16) • The waves travel with the speed of light c where c = 1 √ µoo (17) • Electric field and magnetic field are perpendicular to the direction of propogation. If you take your right hand and curl your fingers from E to B your thumb points in the direction of propogation. Below S points in the direction of propogation. E B S • The magnetic energy per volume stored in the wave is the same as the electric energy per volume stored in the wave uE = 1 2 o E 2 = uB = B2 2µo (18) The total energy per volume in the wave is u = uE + uB (19) 2