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lec07

lec07 - Scott Hughes Massachusetts Institute of Technology...

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Scott Hughes 24 February 2005 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 7: Current, continuity equation, resistance, Ohm’s law. 7.1 Electric current: basic notions The term “electric current” is used to describe the charge per unit time that flows through a region. In cgs units, current is measured in esu/sec, naturally enough. In SI units, current is measured in Coulombs/sec, which is given the name Ampere (or amp). It is very important to know how to convert between these units! Amps are used to describe currents FAR more often than esu/sec: 1 ampere = 2 . 998 × 10 9 esu/sec. Suppose we have a swarm of charges, all with the same charge q . The number density of these charges is some value n (i.e., there are n charges per unit volume). Suppose further that all of these charges are moving with velocity ~u . How much current is flowing through an area ~ A ? To figure this out, we need to calculate how many of the charges pass through the area ~ A in time Δ t . This number is given by the number of charges that fit into the oblique prism sketched below: θ u A u t The number of charges in this prism is its volume, | ~ A || ~u Δ t | cos θ , times the number density. The number of charges that pass through the area in time Δ t is Δ N = n ( | ~u | Δ t )( | ~ A | ) cos θ = n~u · ~ A Δ t . The total charge that passes through is q times this number: Δ q = qn~u · ~ A Δ t , so that the current I = Δ q/ Δ t must be I = qn~u · ~ A . 63

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This form of the current motivates the definition of the current density ~ J : ~ J = qn~u = ρ~u . For this (admittedly artificial) example of all charges streaming in the same direction with the same velocity ~u , the current density is just the charge density ρ nq times that velocity. The current is then given by I = ~ J · ~ A . Caution : bear in mind that the current density is a current per unit area : charge/(time × area). This is hopefully obvious by dimensional analysis: charge density is charge/(length cubed); velocity is length/time; hence current density is charge/(time × length squared). 7.2 Electric current: more details Some of the details assumed above are obviously fairly artificial. We make them more realistic one by one. First, rather than having a single kind of charge that is free to move, a material might have a bunch of different charges that can carry current. For example, ocean water conducts electricity largely because of the freely moving sodium and chlorine ions. Other materials dissolved in the ocean contribute a bit as well. Each variety of charge may have its own charge, number density, and velocity. Let the subscript k label different charge “species” — e.g., k = 1 could refer to electrons, k = 2 to chlorine ions, etc. The total current density comes from combining them all together: ~ J = X k q k n k ~u k = X k ρ k ~u k .
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