1. Current and Resistance (Cont.)
Chapter 27
Current and Resistance (Cont.)
Dr. Jie Zou
PHY 1361

2. Outline A model for electrical conduction (27.3)
Derivation of the drift velocity vd
Conductivity and resistivity in terms of microscopic quantities
Resistance and temperature (27.4)
Electrical power (27.6)
Dr. Jie Zou
PHY 1361

3. A model for electrical conduction
A classical model of electrical conduction in metals: Drude model in 1900
In the absence of an electric field, the conduction electrons move in random directions through the conductor with average speeds ~ 106 m/s. The drift velocity of the free electrons is zero. There is no current in the conductor since there is no net flow of charge.
When an electric field is applied, in addition to the random motion, the free electrons drift slowly (vd ~ 10-4 m/s) in a direction opposite that of the electric field.
(a)
(b)
Dr. Jie Zou
PHY 1361

4. Derivation of the drift velocity, vd, using Drude model
Electric force on an electron: F = qE, where q = -e.
Acceleration of the electron: a = F/me = qE/me.
Define the following:
t = 0: the instant just after one collision has occurred;
t : the instant just before the next collision occurs;
vi: velocity of the electron at t = 0; vf: velocity of the electron at time t.
Apply Newton’s 2nd law: vf = vi + at = vi + (qE/me)t.
Average vf over all possible values of vi and collision time t:
: average time interval between successive collisions = mean free time = relaxation time.
Dr. Jie Zou
PHY 1361

5. Conductivity and resistivity in terms of microscopic quantities
According to Drude model:
Conductivity  = (nq2)/me.
Resistivity:  = 1/ = me/(nq2).
n: charge carrier density = the number of charge carriers per unit volume.
q: the charge on each carrier. For electrons, q=-e.
me: electronic mass.
: mean free time or relaxation time
According to Drude model, conductivity and resistivity do not depend on the strength of the electric field, a feature characteristic of a conductor obeying Ohm’s law.
Mean free path = average distance between collisions:
Dr. Jie Zou
PHY 1361

6. Example 27.5 Electron collisions in a wire
(A) Using the data and results from Example 27.1 and the classical model of electron conduction, estimate the average time interval between collisions for electrons in household copper wiring. (For copper, resistivity  = 1.7 x 10-8 m -see Table 27.1; Charge carrier density n = 8.49 x 1028 electrons/m3).
(B) Assuming that the average speed for free electrons in copper is 1.6 x 106 m/s and using the result from part (A), calculate the mean free path for electrons in copper.
Dr. Jie Zou
PHY 1361

7. Resistance and temperature
Over a limited temperature range, the resistivity of a conductor varies approximately linearly with temperature:  = 0[1 + (T – T0)].
: temperature coefficient of resistivity (see Table 27.1)
Variation of resistance with temperature: R = R0[1 + (T – T0)].
Example 27.6 A Platinum resistance thermometer: A resistance thermometer, which measures temperature by measuring the change in resistance of a conductor, is made from platinum and has a resistance of 50.0  at 20.0°C. When immersed in a vessel containing melting indium, its resistance increases to . Calculate the melting point of the indium.
For a metal, such as copper
For a pure semiconductor, such as silicon
Dr. Jie Zou
PHY 1361

8. Electrical power
Image following a positive charge Q moving clockwise around the circuit from point a through the battery and resistor back to point a.
From a to b through the battery: electric potential energy increases while the chemical potential energy in the battery decreases by the same amount.
From c to d through the resistor: electric potential energy is transformed to the internal energy of the resistor.
When the charge returns to point a, the net result is that some of the chemical energy in the battery has been delivered to the resistor and resides in the resistor as internal energy associated with molecular vibration.
In typical electric circuits, energy is transferred from a source such as a battery, to some device, such as a light bulb.
Dr. Jie Zou
PHY 1361

9. Power: Rate of energy transfer
Rate at which the system loses electric potential energy as the charge Q passes through the resistor:
dU/dt = d(QV)=(dQ/dt) V = I V
Power P, the rate at which energy is delivered to the resistor: P = I V
The above equation can be used to calculate the power delivered by a voltage source to any device carrying a current I and having a potential difference V between its terminals.
For a resistor: P = I V = I2 R = (V)2/R.
Quick Quiz: For the two light bulbs shown, rank the current values carried by each light bulb and their resistance.
Dr. Jie Zou
PHY 1361