Current, Resistance, Power
Electric charges can be made to flow in a conductor. Such a flow of charges is called an electric current.
Consider a small area element A, which might be the cross-sectional area of a conducting wire. If, in a
small time Δt, Δq is the amount of charge that flows through the area element, then we define the current
through that area to be
I=
Δq
.
Δt
The SI unit for current is the Ampere, A. From the definition, we can see that
1 A = 1 C/s.
A current can be related to the motion of the charges creating the current. Consider a conducting wire of
cross sectional area A.
l = vdΔt
charge q
Α
Α
vd
Α
Α
n charge carriers per unit volume
Let n be the number of free charges per unit volume. Assume that each charge carries a charge q and
moves with velocity vd. This velocity is the average velocity of the charges and is called the drift velocity.
Then, in a time Δt, all of the particles inside the volume AvdΔt pass through a specified area element, P.
The number of particles in this volume is nAvdΔt, and the total charge is qnAvdΔt. Thus, the current at
point P is:
I=
Δq
= nqAvd
Δt
The current density is defined to be the current per unit area,
G I
G
j = nˆ = nqvd
A
where n̂ is a unit vector pointing in the direction of the current flow for positive charges. Notice that even
though this is defined by using a conducting wire, the current density is a general quantity that can be
applied to a current whether it is confined to a wire or not.
Resistance
If the current flows easily, we say that the object has a low resistance; on the other hand, if the current
flow is small, then we say that the object has high resistance. We define resistance as the ratio
R=V/I
Where V is in volts and I is in amperes. The SI unit for R is V/C and is called ohm, the symbol for ohms is
Ω.
The other two useful quantities for classifying electrical properties of materials are conductivity and
resistivity. Conductivity is defined to be the ratio of the current density to the electric field:
σ=j/E
If we let the potential drop through a wire be proportional to the length L, i.e. V = EL, then we can relate
conductivity to the resistance as follows:
R=
V EL
EL
L
=
=
=
I
jA σ EA σ A
The resistivity of a material is defined to be ρ = 1 / σ.
The resistance of a sample of length L and uniform cross-sectional area A may be written as
R=
ρL
A
The resistivity of all materials is temperature dependent. For most materials, the relationship is
approximately linear and is given by
ρ = ρ0[1 + α(T – T0)]
where α is the mean temperature coefficient of resistivity appropriate to the temperature range in question
and ρ0 is the resistivity of the material at the reference temperature, T0. Usually, this reference
temperature is 20°C.
Ohm’s Law
Ohm's Law states that the current through a devise is always directly proportional to the applied
voltage, the constant of proportionality is the conductance σ . Because conductance and resistivity are
reciprocals of each other, and because resistance is the measure of the resistivity of a particular object,
we can write Ohm’s law as:
V =IR
A property of an object obeying Ohm’s law is that its resistance will not depend on the amount of current
or voltage applied to it. If the amount of voltage is increased, the amount of current will also increase,
V
⎛
and the resistance will remain the same ⎜ R =
I
⎝
⎞
⎟.
⎠
Power
Recall that in the presence of an electric field, the work done to move a small amount of charge Δq is
ΔW = VΔq.
Since the change in the work is equal to the change in the potential energy, we can use the definition of
power to find:
P=
ΔW
Δq
=V
= VI
Δt
Δt
Since V = IR, we can also express the power as P = I2R or P =
V2
.
R
Notice that if the electrical circuit is attached to a mechanical device, this power will show up as
mechanical work done by the device. If the circuit is attached to an electrical device, the power will be
released as thermal energy heating up the device.
Electromotive Force
Since power is the change in the energy in a circuit per unit of time, we see that in order for the circuit to
continue to provide a steady current there must a source of electrical energy somewhere in the circuit.
ε
This source is called the electromotive force, . Emf has the same units as electrical potential. A typical
example of an emf source is a battery or a generator. The emf of a devise is defined as:
ε = ΔΔWq
We define an ideal emf source as one that maintains a constant potential difference between two points a
and b. An emf source maintains this potential difference, and the work done by the source is the opposite
of the work done by the potential difference on the attached device. Since the emf source does work, by
conservation of energy it must draw the energy from somewhere. Most emf sources use either chemical
energy or mechanical energy.