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Circuits_lectures

# Circuits_lectures - Basic Concept 1 Analogy between the...

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Basic Concept 1: Analogy between the gravitational field and the static electric field m 1 m 2 q 1 q 2 r F = - G m 1 m 2 r 2 ˆ r = - g 12 m 2 ˆ r r F = q 1 q 2 4 0 r 2 ˆ r = E 12 q 2 ˆ r Newton's Law of gravitation Coulomb's Law (the force is always attractive) (attractive or repulsive force) Both are inverse-square, central-force fields. m q Potential Potential energy = mgh h h energy = qEh m q h V Both are conservative fields: The net change in absolute potential around any closed path = 0. r ˆ r r ˆ r r F grav = m r g r F electric = q r E Absolute potential = gh Absolute potential = Eh Ground level (reference height = 0) Ground potential (reference voltage = 0) r F grav r F electric

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This leads to the fundamental Kirchoff's Voltage Law: The sum of the rises and falls of electric potential (voltage) around any closed loop in a circuit = 0. Basic Concept 2: Current and conservation of charge Define current i = q/ t amperes = dq/dt in the infinitesimal limit Circuit loops comprised of sources, resistors, etc. located on the "branches" Surface S q 1 q 2 q 3 r v 1 r v 2 r v 3 q = q 1 + q 2 + q 3 coulombs move through surface S every t seconds
How about current flow leaving a closed surface? S At time t 0 : At time t 0 + t: Q ( t 0 ) = q 4 + q 5 + q 6 + q 7 Q ( t 0 + t ) = q 1 + q 2 + q 3 + q 4 + q 5 i inward = Q ( t 0 + t ) - Q ( t 0 ) t = q 1 + q 2 + q 3 - q 6 - q 7 t For static electric fields, i inward = 0 , i.e., there is no net gain of charge in S over t seconds. Equivalently, i outward = - i inward = 0 . At the green "node" (i.e., the junction of circuit "branches"), this yields the fundamental Kirchoff's Current Law which can be written here as i left + i down + i right + i up = 0 The sum of the electric currents leaving any circuit node = 0. q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 1 q 2 q 3 q 4 q 5 q 6 q 7 i up i down i right i left S S

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Basic Concept 3: Resistance and Ohm's Law Cylindrical bar of conducting material Let a potential of V volts be applied between the flat parallel faces of the bar. Then, the internal electric field is S 1 S 2 r E = V / l in the direction shown, and the force on an individual A electron having the charge – e coulombs is r F e = - e r E . V + Let τ = characteristic time between randomizing collisions of an electron and the nuclei of the atoms within the bar. Note that τ is a statistical average. From Newton's Law, r F e = m e r a e , so we have a e = F e / m e = e V / m e l This yields an average electron "drift velocity" u e = a e = e V / m e l in the rightward direction toward S 2 , the end of the bar having the higher voltage. This electron drift velocity is analogous to the "terminal velocity" of raindrops falling to the ground under the constant downward gravitational force. The raindrops reach a limiting speed despite falling from great heights. In the time period t = l / u e , all of the electrons in the cylinder volume l A are swept rightward through surface S 2 . This represents a total charge movement of q = n electrons / m 3 ( 29 l A m 3 ( 29 - e coulombs / electron ( 29 = - en l A coulombs r F e + l r E + +
Therefore, the rightward current through S 2 is given by i rightward = q t = - e n l A l / u e = - e nAu e = -

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Circuits_lectures - Basic Concept 1 Analogy between the...

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