Day 114

Resistivity
(of an ohmic conductor)

  • R = ρ L/A; where ρ =resistivity of the material (a constant; good conductors have low resistivity); L=length of wire; A=cross-sectional area of wire; ρ measured in Ω m.
  • Resistance is directly related to ρ and L and inversely related to A. Analogy: The longer the artery, the greater the resistance to flow; the wider the artery the less the resistance to flow

Temperature Variation of Resistance

  • As T increases, electrons cannot move as freely; the R increases
  • ρ = ρ0[1 + α (T – T0)]
  • ρ0  = new resistivity at new temp, T0
  • ρ = resistivity at initial temp, T
  • α = temperature coefficient of resistivity (a constant for a material)
  • R = R0[1 + α(T – T0)]

R vs. T

  • Since ρ and R are directly related, we can re-write the previous equation to give:
  • R = R0 [1 + α(T – T0)]
  • For superconductors, R goes virtually to zero below the critical temperature

Ohm’s Law

  • V=IR where V=voltage or potential difference (in V); I=current (in A); R=resistance (in Ω)
  • R is constant for a conductor over a wide range of applied voltages Not all materials obey this law, those are called non-ohmic (vs ohmic)

Ohm’s Law

  • R is determined by the resistor (material, temperature) in the circuit
  • V is determined by the battery source
  • I is determined by V and R
  • Therefore, I=V/R; I is directly related to V and inversely related to R

Mechanical Power and Energy

  • In mechanics, remember that P=W/t; where P=power, W=work, and t=time
  • In electricity, P still = W/t, where P=electrical power, W is electrical energy (since E and W are interchangeable) and T=time

Electrical Power and Energy

  • Since, P=E/t and E=QV (from last unit)
  • Substituting QV for E in P=E/t, yields P=QV/t
  • Since Current, I, = Q/t; this yields P=IV, a new equation for electrical power, still measured in Watts

Power Equations

  • With P=IV and Ohm’s Law, V=IR, combining them yields: P=I2R
  • With P=IV and V=IR, combining them and eliminating I, yields P=V2/R
  • Three different equations can be used to find electrical power, P=IV and P=I2R and P=V2/R
  • You can see that P is positively related to V and I and inversely related to R

Units of Electrical Energy

  • When we open the power bill, we think we are paying for power, but are we really?
  • kWh, the unit on the “power” bill is actually a unit of energy, since “kW” is a power unit times “hours” which is a time unit
  • Power x time = Energy (Work) !
  • Remember, energy has SI units of Joules!

Power Delivered by a Battery

  • If a battery is assumed to be “ideal”, then it has no internal resistance and it delivers all of its voltage to the circuit. If, as in real cases, the battery is not ideal, then it has some internal resistance
  • To find the Voltage consumed by the battery and NOT delivered to the external circuit, use V=Ir, where r is the resistance of the battery. Since there is less voltage to the external circuit, the circuit consumes less power, by P=IV

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