Physics 102
Lab Brief
July 12, 2010
Ohm’s Law
Resistance is Futile
The main task in this lab is to assess Ohm’s law. This is done by studying simple circuits with
resistive elements. We’ll also be evaluating Kirchoff’s Laws and investigating series and parallel
circuits. Additionally, we’ll also examine the difference between Ohmic and non-Ohmic devices.
Ohm formulated his law empirically. Luckily for you, it means that there’s nothing to derive.1
But let’s examen its implications. Let’s start by looking at this form:
I=
V
.
R
(1)
In words, you can think of that relation saying “if I have a circuit with a resistor of resistance R
and a battery applying a constant potential difference of V , then the resulting current drawn from
the power supply flowing through the circuit is I.” The form
V = IR
(2)
lends itself to the interpretation of “voltage drop.” When current flows through a resistive element,
the potential at end of the element where current enters is higher relative to the end at which it
exits. The electrons lose energy as they travel through the resistor (though the rate at which they
flow, the current, remains constant). Hence, in words this form suggests: “a potential drop V will
appear across the terminals of a resistor of resistance R when a current I flows it.”
Ohmic devices are characterized by the linearity their I-V characteristic (plot of I vs. V). While
an I-V characteristic is a more natural plot for investigating circuit elements, we’ll be plotting V vs
I. This has the same information, but its slope results directly in R (whereas the characteristic slope
is 1/R, the conductance). Non-Ohmic devices are those elements whose characteristic is non-linear.
A particular non-Ohmic device of interest to us is the diode. Diodes act like gates which only
open in one direction, so we say they have polarity. As such we call one end the cathode, and the
other the anode (Figure 2). If a potential, above a certain threshold, is applied across the diode,
so that the anode is positive relative to the cathode, the gate opens and current is allowed to flow.
We say that the diode is forward biased. In this state the diode acts like a wire. If, on the other
hand, the cathode’s potential is positive relative to the anode, the gate closes and no current flows
regardless of the value of the applied voltage. Here the diode acts like an open and the state is
referred to as reverse biased. We’ll be using a light emitting diode (LED) which lights up only
when current flows.
We’ll be assessing the validity of Kirchoff’s Rules, so let’s take a moment to review them. The
first rule states that for any loop in a circuit, the sum of the voltage drops across each element is
null. Putting it another way, the sum of the voltages across each element must equal the applied
voltage from some source. The second rule concerns nodes. For each node in a circuit, the sum of
the currents entering a node is equal to the sum of the currents exiting a node (which is essentially
a statement of charge conservation). See the lab documentation for pertinent figures.
Now Kirchoff’s Rules and Ohm’s law yield interesting results when combining resistors.2 Suppose that a potential V is applied across N resistors, with resistances R1 , R2 , . . . , RN , connected
1
There’s a more general form of Ohm’s Law that can be derived, but that requires a working knowledge of electric
fields and details about charge carriers.
2
Disregard that “nothing to derive” comment.
1
Physics 102
Lab Brief
July 12, 2010
Ohm’s Law
in series (so that the same current, I, flows through all, as in Figure 1). Using Kirchoff’s first rule,
we can relate this applied potential to the voltage drop across each resistor:
V = IR1 + IR2 + · · · + IRN = I(R1 + R2 + · · · + RN ).
(3)
Note that because the current is common to all resistors, it can be factored out. It would be nice
if we could replace that mess of resistors with a single, equivalent resistor (Req ) without affecting
any external properties (like the total current drawn from the power supply). If we could do such
a thing, it should obey
V = IReq .
(4)
Comparing our last two equations, you’ll see that we found precisely such a resistor when we
factored out the current:
IA Req = IA (R1 + R2 + · · · + RN )
(5)
⇒ Req = R1 + R2 + · · · + RN .
(6)
The entire network can be replaced with a resistor whose resistance is equal to the sum of the
resistances of the network.3 As far as the power supply knows, these are equivalent. But what if
the network was consisted of resistors connected in parallel (so that a common voltage is across
each resistor, Figure 1 again)? Kirchoff’s Rules still apply, and we haven’t used the second one, so
let’s write that one down:
I0 = I1 + I2 + · · · + IN .
(7)
Here I0 is the total current drawn from the power supply (refer to schematics in lab documentation).
Ohm’s law also still applies, so let’s swap out those currents for voltages and resistances:
V
1
V
V
1
1
I0 =
=V
.
(8)
+
+ ··· +
+
+ ··· +
R1 R2
RN
R1 R2
RN
Now the potential across each resistor is common, so we get to factor that out. If we could replace
this network of parallel resistors with a single Req , it would should be described by
I0 =
V
.
Req
(9)
Notice that the last two equations are identical in form, so equating the total currents (which is
what we meant by equivalent) we can get a form for Req :
1
1
1
=@
V
+
+ ··· +
Req
R1 R2
RN
1
1
1
1
=
+
+ ··· +
.
Req
R1 R2
RN
V
@
(10)
(11)
So if we replace the parallel network with a resistor with a resistance whose reciprocal is equal to
the sum of the reciprocals of the network’s resistances, the power supply would be none the wiser.
Two final things to note about all this tap dancing: the equivalent resistance of a series network is
always larger than any one of the resistances in the network, whereas the equivalent resistance of
a parallel circuit is always less than any of the resistances in the network.
3
Sorry, this definitely looks like it should have been reworded.
2
Physics 102
Lab Brief
July 12, 2010
Ohm’s Law
Wired Up
Acquiring data for this lab is identical to the measurement procedures of the first (Double Zero)
lab. Simply set the voltage applied by the powers supply to some value, measure this with the
voltmeter as well as the current flowing through the circuit. Obtain the resistance by employing a
fitting method on your plots, not a statistical approach on your raw data; do not bother to calculate
the resistance of every voltage-current pair.
We’ll also be determining the resistance of parallel and series circuits. While this may seem a
more complicated system, the equivalent resistance is determined in the same way as before. Treat
series and parallel portions of the circuit as a single resistor; vary the voltage and measure the
resulting total current. The resistance is again determined from your characteristics.
Kirchoff’s Rules can be tested directly in the case of series or parallel circuits. For the former,
test rule one: measure the voltage drop across each resistor and check to see if it adds up as it
should (within uncertainty). For the latter, test rule two: measure the current flowing through the
resistor and see if these sum appropriately (within uncertainty).
For the non-Ohmic devices, don’t bother to fit any curves to the data. Note however, whether
it is linear or not. Because the “knee” in the characteristic for the LED is rather sharp, resolving
it requires some care. Make sure you step through voltage in fine intervals when you are forward
biased in the regime where there is not yet current. Afterward, and also for the reverse biased case,
you can use coarse voltage steps. For this element, use the more standard I-V characteristic for
analysis (in contrast to the V vs. I resistor plots).
Uncertainly
There are two types of measurements you are making in this experiment: voltage and current.
Each is made with a digital meter. The particular meters we’re using claim to provide values that
are within (1% + a unit in the last significant digit) of the true value. If, for example, you are
measuring a potential of 314 V, the resulting uncertainty is .01(314)+1 = 4.14, and would therefore
report the potential as (314 ± 4) V. This is the only source of uncertainty we’ll consider.
Resistances are determined from the slope of a V vs. I graphs4 . Because each data point in
that plot has an uncertainty, the resistance will also. This δR is determined by the range/2 method
(that business with slope extrema). Having the uncertainty in the resistance, we can compare our
determined value with a “known” value, which we take to be the resistance as measured by the
digital meter (this reading also has an uncertainty). If the known value plus/minus its uncertainty
lies within your value plus/minus your uncertainty, you conclude that the values are in agreement.
4
How many times is he going to say that? Must be important...
3
Physics 102
Lab Brief
July 12, 2010
Ohm’s Law
A
A
−
+
R
V
R1
−
+
V
A
R2
−
+
R1
R2
V
Figure 1: Above are the schematics for the circuits used in determining R or Req . The top left schematic
is for the single resistor while the bottom depicts a pair in parallel. The right schematic is for a series
connection. Notice how the meters are set to measure the total current and total potential drop across the
elements of the circuit.
A
A
−
+
R
(a) Forward Bias
−
+
V
V
R
(b) Reverse Bias
Figure 2: The vertex of the triangle of a diode’s schematic symbol (the cathode) indicates the direction
of current flow while the base (anode) visually represents a blockage. A diode is forward biased (allowing
current flow) if its anode is at a higher potential than its cathode (the triangle’s base), and reversed biased
if the opposite is true (precluding current flow). Note also that there is a resistor in series with the diode. If
the diode is to survive more than a few seconds while forward biased, it must be in the circuit.
4