Chapt. 27 Currents and Resistance 205 5. for a single kind of charge carrier of charge q is deFned by v J = nqvV , where n is the density of charge carriers at that point and vV is the velocity of the charge carriers at some point. is a vector and note that it points opposite to vV when q < 0. Imagine a horizontal di±erential bit of surface area dA . The net charge ²owing through this area in a direction an angle θ from the vertical in time Δ t is Δ Q = nq ( V Δ t ) dA cos θ = ( nqvV · d v A )Δ t = ( v J · d v A )Δ t , where ( V Δ t ) dA cos θ is the volume of a parallelepiped that sweeps through dA in Δ t , V Δ t is the length of one side of the parallelepiped, and d v A is the vectorized di±erential area. The di±erential current through d v A is dI = v J · d v A . Integrating over a Fnite surface gives a current: I = i surface v J · d v A .
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