Unformatted text preview: A A E 5 90E 5.3 ELECTRODYNAMICS C h5 – 7 3 A A E 5 90E Introduction Charges are in motion in an electric current. To make these charges flow, one must push on them. How fast they move in response to a given push depends on the nature of the “material”.
q ! F ! v We define (for some substance): Current Density is proportional to a force per unit charge: ! ! ! F J=!"f =!" q V a cu u m A ir / G a s S o lid : M e t a l/ S e m iC o n d u c t o r L iq u id The proportionality constant σ is called Conductivity of a medium. The reciprocal is called Resistivity ρ = 1/σ. In principle, the force driving the charges to produce current could be anything, such as chemical, gravitational, nuclear, etc. Special Case: The velocities of charges are very small, we can neglect mag. forces: ! ! !!! J=! E+v"B =!#f ! ! Ohm’s Law: J = !E ! NOTE: In plasmas, the mag. contribution of f can be significant. ( ) C h5 – 7 4 A A E 5 90E Example Two long cylinders (radii a and b) are separated by material of conductivity σ. If they are maintained at a potential difference V, what current flows from one to the other in a length L?
∆V E σ Electric Field: Voltage: Current: $ˆ s 2" # 0 s !! V = !# E " dl = ! E= ! for L >> r
$ 2% & 0 !! !! I = " J ! da = # " E ! da = #
! "0 ln b a $ 2% & 0 s ! 2% sL = # &0 $L L# = ! "0 L 2$ " 0V ln b a () = V R
C h5 – 7 5 =λ A A E 5 90E “Electromotive Force” Let’s think about what does drive a current in a circuit! ! ! ! ! ! Fel f = fS + E = fS + q ! Source f S confined to one portion of loop (for example: battery), ! ! Electrostatic Force F = q E serves to smooth out flow and communicates the influence of source to distant parts of circuit. What type of sources do we have? Within a source what could be the mechanism to drive charge? Chemical force: Piezoelectric crystal: Thermocouple: Photoelectric cell: battery, mechanical pressure is converted into an electric impulse, temperature gradient of dissimilar metals, photons. Looking at the list of sources, what observation can we draw? The mechanism to drive current (charged particles) in a source are NON–Coulomb forces.
C h5 – 7 6 A A E 5 90E “Electromotive Force” Let’s consider processes in a (mechanical/chemical) battery to examine what’s happening in a source.
! FNC – – + + + + + + + ! ! FC = ! eE C – ! FNC ! EC – – – – – – – + + + + + + + + + + + + + ! ! FC = FNC
! FC ! FNC – ! EC – – – – – – – – – – – – – • Uncharged plates, • “Motor” extracts electrons from the left and transports them to the right, • “Conveyor belt” exerts a ‘NONCoulomb’ force on each electron. • Charge builds up on the plates, • Charges on plates exert a “Coulomb” force on electrons being transported, • FC is in the opposite direction to F N C, • “Coulomb force/field” refers to force and field due to point charges as given by Coulomb’s Law. • Enough charge accumulation on plates generates electric field to balance NON-Coulomb force, • “Motor” cannot move more charges, • Plates are charged to maximum. C h5 – 7 7 A A E 5 90E “Electromotive Force” Important Observations & Conclusions: The potential difference, which can be achieved depends on the balance of forces due to Coulomb and NON–Coulomb forces! We can see that the function of a “motor” is to produce and maintain a charge separation (pull electrons out of the positive plate and push them onto the negative plate). The amount of charge separation is limited by and determined by the strength of the “motor”, for example in a battery by the nature of the chemical reactions.
NON–Coulomb Force !Vsource FNC = EC d = d q
Distance between the plates holding charge Electric Field Strength between the plates C h5 – 7 8 A A E 5 90E Definition of EMF The source performs WORK on the charge to force it to move! DEFINITION: ! dW dq !J $ = V& #C " % Total Work per charge along closed loop: !! !!! !! ! " f " d l = " fS + E " d l = " fS " d l # # # ( ) !! E ! d l = 0 for an electrostatic field! " For ideal source, the potential difference between terminals A and B : no internal resistance: net force on charges is zero: ! =" ! ! E = ! fS !! ! ! B! ! V = ! # E " d l = # fS " d l = " fS " d l = #
B A A ! E + + + + A+ + + + + + + + + ! ! FC = FNC ! FC – ! FNC ! EC
L – – – – –B – – – – – – – –
C h5 – 7 9 A A E 5 90E Definition of EMF NOTE: Historically “EMF” was an abbreviation of “electromotive force”, which is not an appropriate name, since “EMF” is not a force at all but energy input per unit charge! Important: EMF is not a potential difference, although the units are volts, EMF is energy input per unit charge and in principle could be anything: gravitational, nuclear, etc., potential difference is a path integral of electric field made by charges. Only for Ideal Case (ideal source) is the potential difference (maintained by the source) numerically equal to source’s EMF. Real Case: Due to internal resistance in sources, the potential difference between the terminals (outside source loop) will be less than the numerical value of EMF.
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- Fall '09
- Electromotive Force, Magnetic Field, Potential difference, Electric charge, FNC