UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
APSC 178: Electricity and Magnetism
MODULE 5.0
Current, Resistance, and Electromotive Force
Readings: Young and Freedman, Chapter 25
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
5.1 Current
A current is any motion of charge from one region to another.
In the electrostatic situation, the electric field is zero everywhere. However, this does not mean that the
charges are at rest. In conducting materials like copper charges are free to move about within the material.
These charges move randomly in all directions, so there is no net flow of charge in any direction and hence,
no current.

If a steady
electric
field
E
is established inside the conductor, charges are subjected to a steady force

F  qE . However, charges also undergo frequent collisions with the massive ions in the material. In each
such collision the particles’ direction of motion undergoes a random change. The net effect of the electric
field is that in addition to the random motion of the charges, there is a slow net motion of drift of moving
charged particles in the direction of the electric force. This motion is described in terms of a drift velocity.
Hence, there is a net current in the conductor.
We define the current, denoted by I, to be in the direction in which there is a flow of positive charge. Thus,
we describe currents as though they consisted entirely of positive charge flow. This choice or convention for
the direction of current flow is called conventional current.
We define the current through the cross-sectional area A to be the net charge flowing though the area per unit
time. Thus, if a net charge dQ flows through an area in a time dt, the current I through the area is
The SI unit for current is the ampere.
Current, Drift Velocity, and Current Density
We can express current in terms of the drift velocity of moving charges. Suppose there are n moving charged
particles per unit volume moving at some drift velocity vd. If each particle has a charge q, the charge dQ that
flows through an area A during a time dt is
This corresponds to a current:
and a current per unit cross-sectional area is called the current density J:
The SI units of current density are amperes per square metre.
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau

We can also define a vector current density J that includes the direction of drift velocity:
Example 1, Current density and drift velocity in a wire: An 18-gauge copper wire has a nominal diameter of
1.02mm. This wire carries a constant current of 1.67A to a 200 W lamp. This density of free electrons is
8.5×1028 electrons per cubic metre. Find the magnitudes of (a) the current density and (b) the drift velocity.
5.2 Resistivity


For many materials, J is directly proportional to E (Ohm’s Law). We define the resistivity ρ of a material as
the ratio of the magnitudes of the electric field and current density:
The reciprocal of resistivity is conductivity in units of (Ω·m)-1.
Material
Silver
Nichrome
Silicon
Glass
Wood
Resistivity (Ω·m)
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Classification
UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
5.3 Resistance
Consider a conductor with uniform cross-section A and length L. The current density J is uniform over the
cross-section and the electric field E is constant along the length.
The total current through the wire is: _____________________
The potential difference between the two ends is:_________________________
The ratio of V to I for a particular conductor is called the resistance R=V/I.
Example 1, Q25.63: A material of resistivity ρ is formed into a solid, truncated cone of height h and radii r1
and r2 at either end. (a) Calculate the resistance of the cone between the two flat end faces. (b) Show that
your result agrees with the expected result when r1 = r2.
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
5.4 Electromotive Force and Circuits
For a conductor to have a steady current, it must be part of a path that forms a closed loop or complete
circuit. If the path is not a closed loop, then charge quickly piles up on either end of the path and current
stops flowing.
To maintain charge flow (current) in a closed path, there must a source of potential energy that increases the
potential energy of charges. The influence that makes current flow from lower to higher potential is called
electromotive force (emf). Electromotive force is not a force, and has SI units of V.
Schematic diagram of a source of emf in an “open-circuit” situation. The electric field force and the nonelectrostatic force are shown for a positive charge q.
Schematic diagram of a source of emf in a complete circuit. The electric field force and the non-electrostatic
force are shown for a positive charge q. The current is in the direction from a to b in the external circuit and
from b to a within the source.
Internal Resistance
Real sources of emf in a circuit have internal resistance due to the motion of charge through the material of
the source. When current flows through a source from the negative terminal b to the positive terminal a, the
potential difference between the terminals is:
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
Why is the terminal voltage less than the emf?
Symbols for Circuit Diagrams
Conductor with negligible resistance:
Resistor:
Source of emf:
Source of emf with internal resistance:
Voltmeter:
Ammeter:
Example 1, Q 25.69: The potential difference across the terminals of a battery is 8.4V when there is a current
of 1.50A in the battery from the negative to the positive terminal. When the current is 3.50V in the reverse
direction, the potential difference becomes 9.4V. (a) What is the internal resistance of the battery? (b) What
is the emf of the battery?
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
Example 2, Using voltmeters and ammeters: What are the voltmeter and ammeter readings in the situations
shown below.
Potential Changes around a Circuit
The net change in potential energy for a charge q making a round trip around a closed circuit must be zero:
Potential gains are associated with:__________________
Potential drops are associated with:___________________
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
Example 3: What is the potential Vad in the circuit below? (b) What is the terminal voltage of the 4.00V
battery? (c) A battery with emf 10.30V and internal resistance 0.50Ω is inserted into the circuit at d, with its
negative terminal connected to the negative terminal of the 8.00V battery. What is the difference of potential
Vbc between the terminals of the 4.00V battery now?
5.5 Energy and Power in Electric Circuits
In electric circuits we are interested in the rate at which energy is either delivered to or extracted from a
circuit element. If the current through the element is I, then in a time interval dt, the amount of charge that
has passed through the element is:
The potential energy change for this amount of charge is:
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
Dividing this expression by dt yields the rate at which energy is transferred, or power P:
Power Input to a Pure Resistor
The potential difference across a resistor is:
Thus, the electrical power delivered to the resistor by the circuit is:
Here, the potential at a is always higher than at b (potential is lost as a charges moves through the resistor).
This means that the rate of transfer of energy is into the device.
Power Output of a Source
Energy is delivered to the external circuit at a rate:
For a source with internal resistance yields a power output of:
The emf term is the rate at which work is done on the circulating charges.
The internal resistance term is the energy dissipated in the source.
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
Power Input to a Source
Suppose a source is connected to another source, with an emf larger than that of the source. Then the current
is pushed backward through the upper source. This reversal of current results in a potential:
We have a power input into the source of:
A common example of this effect is battery chargers.
Example 1, Power input and output in a complete circuit: (a) Find the rate of energy conversion and the rate
of dissipation in the battery and the net output power of the battery. (b) If the 4Ω resistor is replaced with an
8Ω resistor, how would this affect the electrical power dissipated in the resistor?
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