Chapter 25 – Current, Resistance, and Electromotive Force
I. Introduction: We will now look at the situation where charges are in motion - electrodynamics. The major difference between the static and dynamic cases is that E = 0 inside conductors for the static case, but E 0 inside conductors for the dynamic case. And, if E 0, then charges in the conductor feel a force (F = qE) and move in response to that force.
II. Examine what happens to the electrons within a conductor:
A. static case: E = 0
There still exists a thermal motion, and the average position of the charges does not change (even though electrons will be moving at approximately 105 m/s between collisions when the temperature is 300 K.)
B. dynamic case: E 0
When the electric field is present, the movement of the charges caused by the electric field is superimposed on the thermal motion, and there is a net motion of the charges (in the direction of the force exerted on the charges due to the electric field – in the same direction as the E-field for positive charges and in the opposite direction of the E-field for negative charges). The average speed at which the charges move is called the drift velocity, vd.
The motion of electrons (negative charge) in a particular direction can be replaced equivalently by the motion of positive charges in the opposite direction. In our discussion of charge motion we will deal primarily with motion of positive charges.
III. How do we produce the electric field in the conductor?
When positive and negative charges are separated, an electric field is produced pointing from the positive to the negative charge. Any device that supplies the energy to cause the charges to separate is referred to as an EMF (electromotive force). The EMF produces the necessary electric field to cause the charges to move, e.g., battery (chemical energy), generator (mechanical energy), etc. Below is shown a conductor (wire) connected between the ends of a battery. The electric field produced by the battery causes the charges to move, and this motion of charges is called an electric current.
IV. Electric Current, I
A. Imagine a section of the wire above, , with a cross-sectional area A and with charges flowing with a velocity vd . The direction of current flow is taken as the direction in which positive charges flow (even though in wires the negative charges, electrons, are the ones flowing).
1. The electric current is defined as the amount of charge crossing an imaginary boundary in the wire per unit time:
electric current =
units:
2. The average electric current,
3. The instantaneous value of the electric current, I
4. For a constant current:
5. The value of the electric current in a wire is the same no matter how the cross-sectional area of the wire might change.
B. Another quantity that is closely related to current and does depend on the cross sectional area of the wire is current density J. The current density is the electric current per cross-sectional area, that is,
J =
C. When the current density is constant: (across the cross-section)
D. How are the electric current, I, the current density, J, and the drift velocity, vd , related?
Let n = number of free charges per unit volume
q = value of the charge
A = cross-sectional area of the wire
In a time dt, the charges travel a distance dx , then
Example: What is the drift velocity of the electrons in a 2 mm diameter copper wire carrying a current of 1 A? Take n = 8.5 x 1028 electrons per cubic meter for copper. Before making the calculation, what is your guess?
V. Resistance and Resistivity
A. Because collisions take place as the electrons move through the wires, we can say that the wires are producing an impeding effect to the flow of those charges. This impeding effect due to the collisions is called electrical resistance.
1. Relationship between .
2. Some values of resistivity, r, and conductivity, s = 1/r, at 20oC.
copper: r = 1.7 x 10-8 ohm m s = 5.9 x 107 (ohm m)-1
carbon: r = 3.5 x 10-5 ohm m s = 2.9 x 104 (ohm m)-1
glass: r ~ 1012 ohm m s ~ 10-12 (ohm m)-1
B. We notice that the resistivity depends on temperature. So, how does the resistivity of various types of materials change when the temperature is changed? Why?
1. conductors: r increases as T increases
2. semiconductors: r decreases as T increases
3. insulators: r decreases as T increases
C. What is the quantitative relationship between the change in resistivity and the change in temperature? For relatively small changes in temperature, what happens?
Values of the temperature coefficient of resistivity, a, is a characteristic of the material, e.g.,
copper: a = 3.9 x 10-3 (oC)-1 , at 20oC
carbon: a = -0.5 x 10-3 (oC)-1 , at 20oC
germanium: a = -48 x 10-3 (oC)-1 , at 20oC
D. In electrical circuits, E and J are not measured, but V (voltage) and I (current) are. Let’s find the relationship between these quantities.
Consider a length of the wire where an electric field E and the resulting current density J are present. Find the potential difference between points A and B, that is, find VAB = VA – VB .
1. Define resistance:
units: ohms or W
2. For a wire with a uniform cross-sectional area A and length L (most wires).
R = r.
3. A device that is designed to have resistance is called a resistor and uses the following symbol in circuit diagrams.
4. The relationship between V and I above can be written as
which is the basic relationship between the voltage across and the current through a resistor. When V is proportional to I, this relationship is referred to as Ohm’s Law.
E. Look at resistance and voltages for various lengths of copper wire.
1. Take a 1 m length of copper wire with a diameter of 2 mm carrying a current of 2 A at a temperature of 20oC. What is the resistance of the wire, and what is the voltage between the ends of the wire?
2. Suppose the temperature of the above wire is increased to 220oC. What is the new resistance?
3. What is a superconductor?
4. Here’s an example where the wire does not have a uniform cross-section. Find the resistance of a truncated cone made from a material whose resistivity is r, whose radii are a and b, and whose length is L.
VI. EMF (electromotive force) - the things that produce the electric field that drives the charges in an electrical circuit. Examples: battery, generator, power supply, etc.
Consider a battery as an example. Find the voltage across the battery.
A. Battery (emf) not connected to anything (battery on “open circuit”)
VAB = VBA =
B. Battery connected to a circuit - a current flow exists in the circuit due to the battery.
Current flows through the wires, and the same current flows through the battery. With the flow come collisions that impede the flow through the battery. This is referred to as the internal resistance, r, of the battery.
Now, what is the potential difference, VAB ?
C. Battery connected to the circuit, but current is forced to flow "backward" through the battery.
In this case the battery is being "charged." The chemical reaction is being reversed.
What is VAB ?
D. Circuit symbol for a battery.
VII. Energy and Power
A. The work required to move a charge dq through a potential difference V is dW = (dq)V.
Then, the power needed to accomplish this is
Power = P = = , since I = ,
Power P = IV.
This is an expression that can be used for any circuit element. The units are amps x volts (A v) = Watt (W).
B. Look at different situations:
1. Power delivered to a resistor or dissipated in a resistor. (Rate at which energy is supplied to or lost in a resistor.)
2. Power delivered by a battery. (Rate at which energy is supplied by the battery.)
3. Power delivered to a battery. (Rate at which energy is supplied to the battery in order to charge it.)
VIII. Example of voltage and current calculations for a battery connected to a simple circuit.
A. The battery has an emf of 1.50 volts and an internal resistance of 2.00 ohms. We also have an 8.00 ohm resistor, an ideal ammeter, and an ideal voltmeter. The diagram shows the battery.
1. What is the “open circuit” voltage of the battery?
2. The 8.00 ohm resistor is connected to the battery.
a. Find the current flowing past points A, B, and C.
b. Find the voltage across the battery.
c. Find the voltage across the resistor.
d. Find the power delivered by the battery.
e. Find the power dissipated in the resistor.
f. Find the “short circuit” current in the battery.
3. Circuit with multiple resistors and batteries.
a. Find the magnitude and direction of the current in the circuit.
b. Find the terminal voltage VAB of the 10.0 volt battery.
c. Find VAC.
B. Using voltmeters and ammeters to read voltage and current values.
1. The Voltmeter
a. gives the value of voltage between any two points in the circuit (in parallel)
b. ideally has an infinite resistance so that no current passes through it (ideally does not affect the circuit)
2. The Ammeter
a. gives the value of current flowing past a specific spot in the circuit (in series) – you cut the circuit where you want to know the current and insert the ammeter
b. ideally has no resistance (ideally does not affect the circuit)
3. Where do we connect the voltmeter and where do we place the ammeter to make the following measurements in the circuit shown?
a. VAB
b. VAD
c. VCD
d. IC
.
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