notes25 2317 - ECE 2317 ECE Applied Electricity and...

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Prof. David R. Jackson ECE Dept. Spring 2011 Notes 25 ECE 2317 ECE 2317 Applied Electricity and Magnetism Applied Electricity and Magnetism
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KCL Law KCL Law 1 0 0 N n n tot in i i = = = Wires meet at a “node” i 2 i 3 i N i 4 i 1
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KCL Law (cont.) KCL Law (cont.) A C +Q tot in dv i C dt = ground C is the stray capacitance between the “node” and ground. + - v A = surface area of “node.”
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KCL Law (cont.) KCL Law (cont.) 1) In “steady state” (no time change) 2) As A (area of node) 0 , C 0 0 0 tot in dv dt i = = 0 tot in i = tot in dv i C dt = Two cases for which the KCL law is valid:
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KCL Law (cont.) KCL Law (cont.) In general, the KCL will be accurate if the size of the “node” is small compared with a wavelength λ 0 . node 8 0 2.99792458 10 c f f λ × = = f λ 60 Hz 5000 [km] 1 kHz 300 [km] 1 MHz 300 [m] 1 GHz 30 [cm]
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KCL Law (Differential Form) KCL Law (Differential Form) 1 0 0 N tot n in n i i = = = or To obtain the differential form for static (D.C.) currents, start with the definition of divergence: 0 J ∇⋅ = 0 1 ˆ lim V S J J n dS V ∇⋅ Ñ ˆ 0 tot out S tot in J n dS i i = = - = Ñ J V (valid for D.C. currents) For D.C. : (circuit form) For the right-hand side: Hence
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Important Current Formulas Important Current Formulas Ohm’s Law J E σ = Charge-Current Formula v J v ρ = (This is an experimental law that was introduced earlier in the semester.) (This was derived earlier in the semester.) These two formulas hold in general (not only at DC).
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