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2102

Jonathan Dowling PPhhyyssicicss 22110022 LLeeccttuurree 1188 CChh3300:: IInndduuccttoorrss && IInndduuccttaannccee IIII

Nikolai FFaarraaddaayy’’ss LLaaww • A varying magnetic creates an induced n B EMF • Definition of is similar to definition of dA 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 , exhibited as a current. • Recall that a current flows in a conductor because of the on charges produced by dA an electric . • Hence, a time varying r d! magnetic flux must induce an E #dsr = " B ! $ • But the electric C dt would be closed!!?? What about Another of ’s equations! difference ΔV=∫E•ds? To decide SIGN of flux, use right hand rule: fingers around loop C, thumb indicates direction for dA. r d! $ E #dsr = " B C dt EExxaammpplele

A long 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 produces a 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 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. → 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 Units : [L] = ! H (Henry) (1799-1878) ““SSeelflf””--IInndduuccttaannccee ooff aa ssoolelennooidid • Solenoid of cross-sectional area A, 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 = “” = µ NAn = µ A 0 0 l di EMF = !L dt EExxaammpplele i • The current in a 10 H 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