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Resistance

1. Current

   I ≡ dq / dt

   is measured in Amperes (1 A ≡ 1 C / s).

   Its direction is conventionally the direction of movement positive charge carriers would have, from positive to negative potential, even though the charge carriers are actually negative. So electrons move in the opposite direction to the current!

   Viewed as charges "drifting" down a wire, I = ρ v_drift A, where A is the cross-sectional area of the wire.
2. We define the current density J ≡ I / A. For many materials of interest,

   J = σ E = E / ρ

   where here σ is the conductivity of the conductor and ρ is now the resistivity. Note that E is no longer zero inside the conductor: it is the cause of the current. Assuming a uniform electric field, we have

   J = ΔV / (ρ l)

   and Ohm's law:

   ΔV = I (ρ l / A)

   ≡ I R

   where R is measured in Ω (Ohms).
3. The resistivity is a function of temperature; expanding in a Taylor series, and truncating after the linear term, we have

   ρ(T) = ρ₀ + dρ/dT (T₀) * (T - T₀)

   ≡ ρ₀ (1 + α (T - T₀))

   where α is the temperature coefficient of resistivity.
4. A resistor dissipates electrical power at the rate of

   P = dU/dt

   = d(q ΔV)/dt

   = I ΔV.

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©2010, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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