This is an expanded version of a previous guide. The present guide includes additional information about resistivity and power.
About conventions regarding signs:
Conventional current is positive current. We now know--unlike
in Ben Franklin's day--that electric current is due to the movement of
electrons; hence, electrical current is negative current. Ben Franklin
didn't know that and assumed the opposite. His convention stuck and so
now we conventionally talk about current as the movement of positive
charge even though we know otherwise. The physics still works out as
long as we make the direction of positive current opposite that of
electron current. Generally, when we just say current without an
adjective, we mean conventional (positive) current. Note that inside a
battery, the direction of positive current is from the negative to the
positive terminal while in the circuit outside the battery, positive
current goes from positive to negative.
symbol I represents conventional current. That doesn't mean, however, that you
can't get negative values for I. If you solve a
circuit problem and I comes out negative, that may just mean that
the conventional current is in the opposite direction as you thought.
You'll find that the
textbook often uses the symbol V to represent potential difference
(sometimes called voltage for short). However,
it makes more sense to represent potential difference with a symbol that
means change, namely ΔV. This
represents the difference of potential between two points in a circuit.
If the points are a and b, then another way to write this
is ΔV = Vba = Vb - Va.
This makes it clear that potential difference can be positive or
negative. Note that inside a battery, the potential rises from the
negative to the positive terminal while in the circuit outside the
battery, potential falls from positive to negative.
Electrical resistance is always positive.
The definition of electrical resistance is R = -ΔVr/I. How do you get a positive number out of that? Well, when positive
current passes through a resistor, the change of potential across the
resistor is negative. So -ΔVr is positive.
symbol P for power represents a positive number. For a
battery, then, P would represent a gain while for a resistor, P would
represent a loss.
||I = Q/t
||C/s or A
||Conventional current is taken
to be positive current.
||ΔV or Vba
||ΔV = ΔUel /Q
||Potential difference is the
change in electrical potential energy per unit charge as charge Q moves from point to point in a circuit. This is referred to as
voltage and represented by V in the textbook under the
assumption that the lower potential is always 0. In your problem
solutions and lab work, avoid use of the naked V symbol.
||R = -ΔVr /I
||Since ΔVr across a
resistor is negative (from + to -) and current is positive (from
+ to -), the quantity R = -ΔVr /I is always positive.
||R = ρL/A
||The resistivity characterizes
the resistance of a wire of length L and cross-sectional area A.
The resistivity depends on the material and may also be
influenced by temperature.
||ρ = ρo[1 + α(T - To)]
||The temperature coefficient of
resistivity characterizes how the resistivity of a material
depends on temperature. In the formula, ρo is the
resistivity at some reference temperature, and ρ is the
resistivity at a temperature ΔT different from the reference
||P = |ΔUel |/Δt
||The absolute value of the
change of electrical potential energy per unit time is the
power. Letting P always represent a positive quantity is
conventional. In a battery where electrical potential energy is
gained from the negative to positive terminal, P would represent
a power gain. In a resistor where electrical potential energy
is lost from the positive to negative side, P would represent a
power loss. This is sometimes referred to as power dissipation.
||The fact that charge is
conserved leads to the conclusion that the current is the same
in all parts of a single loop circuit. From the definition of
current, I = Q/Δt. For a given Δt, equal charge means
||The total change in electric
potential energy ΔUel around a circuit is 0.
All the energy produced by the battery is used in the circuit.
Now ΔUel = QΔV, where ΔV is the difference potential
around the circuit. But the difference of potential in returning
to the same point must be 0. Another way of saying this is that
the algebraic sum of all the potential differences around a
circuit must be 0. This is simply an expression of conservation
ΔVr is proportional to I
ΔVr = -RI, where R is constant
|Note the difference in the way
we express this law from that in the text. Again, we emphasize
that there is a difference of potential. We also
emphasize the proportionality between the difference of
potential across a resistor and the current in the resistor. The
constant of proportionality is -R. The negative sign is needed,
because ΔVr is negative for positive current in a resistor.
Note that Ohm's Law isn't obeyed for many circuit components and
hence, doesn't have the stature of, say, the Law of Gravitation
or Newton's Laws.
|Power production in a battery
||Pb = I |ΔVb|
||The power gain in a battery is
the increase of potential across the terminals multiplied by the
|Power dissipated in a resistor
Pb = |ΔVr|2/R
|The power dissipated in a
resistor can be expressed either as the square of the potential difference
across the resistor divided by the resistance or as the square
of the current in the resistor multiplied by the resistance. A
derivation of this is given below.
More about power in a simple
circuit of a battery and resistor shown above.
Positive charge Q passing from D to A in the battery gains electrical
potential energy ,
where is the difference of potential across the battery. The power gain Pb is:
Positive charge Q passing from
B to C in the resistor loses electrical potential energy ,
where is the difference of potential across the resistor. The power loss Pr is:
Here is an alternative
derivation for Pr: