Physics 2102 Jonathan Dowling PPhhyyssicicss 22110022 LLeeccttuurree 1188 CChh3300:: IInndduuccttoorrss && IInndduuccttaannccee IIII Nikolai Tesla FFaarraaddaayy’’ss LLaaww • A time varying magnetic FLUX creates an induced n B EMF • Definition of magnetic flux is similar to definition of dA electric flux r r ! = B "dA B # S • Take note of the MINUS sign!! • The induced EMF acts in such a d! B way that it OPPOSES the EMF = " change in magnetic flux dt (“Lenz’s Law”). AAnnootthheerr ffoorrmmuulalattioionn ooff FFaarraaddaayy’’ss LLaaww • We saw that a time varying magnetic FLUX creates an n B induced EMF in a wire, exhibited as a current. • Recall that a current flows in a conductor because of the forces on charges produced by dA an electric field. • Hence, a time varying r d! magnetic flux must induce an E #dsr = " B ELECTRIC FIELD! $ • But the electric field line C dt would be closed!!?? What about electric potential Another of Maxwell’s equations! difference ΔV=∫E•ds? To decide SIGN of flux, use right hand rule: curl fingers around loop C, thumb indicates direction for dA. r d! $ E #dsr = " B C dt EExxaammpplele A long solenoid has a circular cross-section of radius R. R The current through the solenoid is increasing at a steady rate di/dt. Compute the electric field as a function of the distance r from the axis of the solenoid. The electric current produces a magnetic field B=µ0ni, which changes with time, and produces an electric field.The magnetic flux through circular disks Φ=∫BdA is related to the circulation of the electric field on the circumference ∫Eds. First, let’s look at r < R: Next, let’s look at r > R: magnetic field lines dB E (2! r) = (! r 2 ) dB dt E (2!r) = (!R2 ) di dt = (! r 2 )µ n 0 dt µ n di µ n di R2 E = 0 r E = 0 2 dt 2 dt r electric field lines EExxaammpplele ((ccoonnttininuueedd)) µ n di µ n di R2 E = 0 r E = 0 2 dt 2 dt r E(r) magnetic field lines r r = R electric field lines SSuummmmaarryy Two versions of Faradays’ law: – A varying magnetic flux produces an EMF: d! EMF = " B dt – A varying magnetic flux produces an electric field: r d! $ E #dsr = " B C dt IInndduuccttoorrss:: SSoolelennooididss Inductors are with respect to the magnetic field what capacitors are with respect to the electric field. They “pack a lot of field in a small region”. Also, the higher the current, the higher the magnetic field they produce. Capacitance → how much potential for a given charge: Q=CV Inductance → how much magnetic flux for a given current: Φ=Li di Using Faraday’s law: EMF = !L dt Tesla " m2 Joseph Henry Units : [L] = ! H (Henry) (1799-1878) Ampere ““SSeelflf””--IInndduuccttaannccee ooff aa ssoolelennooidid • Solenoid of cross-sectional area A, length l, total number of turns N, turns per unit i length n • Field inside solenoid = µ0 n i • Field outside ~ 0 ! B = NAB = NAµ0ni = Li N 2 L = “inductance” = µ NAn = µ A 0 0 l di EMF = !L dt EExxaammpplele i • The current in a 10 H inductor is decreasing at a steady rate of 5 A/s. • If the current is as shown at some instant in time, what is the magnitude and direction of the induced EMF? • Magnitude = (10 H)(5 A/s) = 50 V • Current is decreasing (a) 50 V • Induced emf must be in a direction that OPPOSES this change. (b) 50 V • So, induced emf must be in same direction as current.