While originally used to describe metallic wires, the relation known as Ohm's law has proven useful for describing the behavior of many materials. This principle was discovered in 1827 by Georg S. Ohm (1787 - 1854) and is the basis of many essential facts in practical electricity. It relates the current $I$ through a component in an electrical circuit to the potential difference $V$ across that element and to the resistance $R$ of the component as

Current is typically measured in amperes ($\text{A}$), potential difference is measured in volts ($\text{V}$), and resistance is measured in ohms ($\Omega$).

There exists a large class of materials where the value of $R$ is found experimentally to be independent of $V$ and $I$. In such a case, Eq. (1) is a very simple linear relation between the current through and the voltage across the element. Such elements are ordinarily called resistors, symbolized as shown in Fig. 1. They are said to be linear or ohmic devices.

Figure 1: Symbol for a resistor

Today, there are many components or chips that rely on semiconducting materials, which cannot be assigned a static value for resistance. The entirety of digital computing is made possible due to such compounds, so substances that don't obey ohm's law can be as important as those that do. Furthermore, some substances may have resistances that depend on other physical properties, such as their temperature, illumination, or strain. These are what physicists typically end up needing knowledge of electronics for, as turning physical properties into electrical signals makes it much easier to record and measure important physical processes.

• Familiarize yourself with the concepts of electrical potential, current, and resistance, and explore a model of how they relate to one another.
• Gain experience with instrumentation and basic circuit building, and develop intuition on how measurement devices work.
• Learn about the important voltage divider circuit and how such a circuit can be used to create a sensor.

In past labs, you have seen a number of circuit diagrams. These diagrams differ from many other illustrations used in physics because they do not depict the physical state of the circuit. Instead, they are a blueprint for how to electrically connect components. We will look a bit more carefully at such diagrams today.

To prevent blowing fuses in power supplies and meters, please use the following steps when connecting circuits:

• Turn off the power supply.
• Connect the circuit. For clarity, connect series loops first, then add sections in parallel (e.g., voltmeters). Follow wiring diagrams point by point as you connect the circuit.
• Select the meter function (current, voltage, resistance, etc.).
• Turn on the power supply and increase the voltage slowly.

Many resistors are marked with colored bands which indicate their resistance values and tolerance. The bands are to be read starting closest to the end of the resistor as shown in Fig. 2.

Figure 2: Resistor color code

The digit values represented by each color are shown in Table 1. The tolerance values represented by each color are shown in Table 2.

For example, if the colors are red, orange, yellow and silver, then the resistance is $R = 23\times 10^4\text{ \(\Omega\)} \pm 10\%$ or $230\text{ k\(\Omega\)}$. 

If there were three bands that were brown, black, and brown, then the resistance would be $R = 10\times 10^1 \text{ \(\Omega\)} \pm 20\%$ or $100\text{ \(\Omega\)}$.

One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.

Making electrical measurements is a bit different than most of the other types of measurements we've done. Using a ruler to measure a length won't change something's length, nor will weighing an object with a scale change its mass. But, using a multimeter can change how a circuit behaves if we're not careful. Thus, if you're seeing something on a meter that you don't expect, double check that the correct terminals are attached to the circuit.

Take a look at the generic multimeter shown in Fig. 3. One wire should always be connected to the common (COM) terminal, and the other will be attached to one of the following:

• “$\mathrm{V}/\Omega$“ for measuring any voltage or any resistance;
• “$200 \text{ mA}$” for measuring a small current; or
• “$10\text{ A}$” for measuring a large current.

For measuring voltages, you'll connect the meter to either side of a component (i.e. in parallel).

For measuring currents, you'll connect the meter in-line with the component (i.e. in series).

For measuring the resistance (of a static device like a resistor), you'll connect the meter directly to either side of the component (with nothing else connected in the circuit).

Finally, you'll need to set the dial to what you want to measure. The tilde ~ indicates an AC value (not used today), whereas the straight & dashed line combination indicate DC values.

Figure 3: Depiction of a generic multimeter. Your version may have a different layout, multiple settings for each measurement type, or other features.

Using a resistor value between $1\text{ k\(\Omega\)}$ and $10\text{ k\(\Omega\)}$, set up the circuit in Fig. 4.

Figure 4: Current measurement

• Plug the black test lead into the “COM” jack on the multimeter and the red test lead into the “$\text{mA}$” jack. Set the function of the digital multimeter to be an ammeter.
• Set the large current limit knob on the power supply to 0.2 A and set the smaller current limit knob fully clockwise.
• Turn on the power supply and observe the meter. If the reading is negative, you may have connected the circuit differently from the diagram. Check it.
• Adjust the power supply and observe and record the range of currents. Turn off the power supply.

Using the same circuit as above with the power off, add a voltmeter across $R$ as in Fig. 5.

Figure 5: Simultaneous measurement of current and voltage

• Since the maximum voltage possible from the power supply is $50\text{ V}$, set this second meter to read DC volts.
• Plug the black test lead into the “COM” jack and the red test lead into the “$\text{V}$” jack.
• Turn on the power supply. If the reading is negative, check the circuit.
• Adjust the power supply and observe the range of voltage available.

Now that you have a bit of practice with the meters, you'll use them to test the behavior of a few different conductive elements. For each element, you will vary the power supply voltage and measure the voltage across the element along with the current through it. If a material is Ohmic, then a plot of the voltage versus current ($V$ vs. $I$) should be linear; if it is not Ohmic, then the curve will have a different shape. As you test, remember that you can reverse the connections from the power supply to produce negative voltage differences to see if the element behaves differently when the direction of the current flow is reversed.

Produce voltage versus current plots for the following elements:

A resistor

A light bulb

An LED (Light Emitting Diode) (This is actually an LED in series with a resistor, which helps keep us from damaging the diode)

In your lab notebook, show your plot for each element and include a brief description of what you observe for the ones that emit light. (If your resistor emits light, you've done something terribly wrong.) Which elements are Ohmic and which are not?

Next, you'll practice connecting multiple elements together in different configurations.

• Choose two resistors, measure their resistances individually, and then connect them in series as in Fig. 6.
• Using the resistance values for each resistor measured earlier, calculate the expected total series resistance.
• For one setting of the power supply, measure $V$ and $I$ and determine the total resistance.

Figure 6: Resistors in series

• Connect your resistors and the meters in the circuit shown in Fig. 7.
• For one setting of the power supply, measure $V$ and $I$ and determine the total resistance.

Figure 7: Resistors in parallel

Now we'll do something that is ubiquitous in electronics: dividing voltage. In some cases, this is done so that different things can be powered by a single battery. In other instances, voltage dividers form the basis of sensors that connect physical properties (like position) into electrical properties (like voltage).

Here we'll introduce the notation of a voltage input $V_{\text{in}}$ and a voltage output $V_{\text{out}}$. For the purposes of this class $V_{\text{in}}$ will usually denote a voltage from a power supply, and $V_{\text{out}}$ will denote where to measure a voltage to determine the effect of the circuitry.

Connect the circuit shown in Fig. 8. Be sure that both meters are on the DC voltage setting. This circuit is useful when a fixed fraction of some input voltage is needed.

Figure 8: Voltage divider

From Ohm's law and how resistors add in series, it can be shown that

For one power supply setting, measure the input and output voltages to test the above relation for your circuit.

Replacing $R_2$ with an element whose resistance depends on some physical quantity (e.g. temperature, pressure, light) is a way of making a basic sensor.

 A variable voltage divider can be made by replacing the fixed resistors with a length of resistance wire and a sliding contact as shown in Fig. 9.

Figure 9a: Slide-wire potentiometer

Figure 9b: Schematic circuit diagram of potentiometer

For this setup, you'll test the effect changing the sliding contact has on the output voltage $V_{\text{out}}$. If the wire is uniformly made, then the resistance across a portion of it is proportional to length. Using this with Eq. (2), we can find the relation:

Keeping $V_{\text{in}}$ constant, test this relationship and record your results (along with a plot) in your notebook.

Now that you've seen how a fixed voltage divider and an adjustable voltage divider work, it isn't too large a leap to translate these concepts into a practical circuit. If one of the resistors is fixed and the other one is allowed to vary (either from a mechanical response – like turning a knob – or from a physical response – like reacting to light or pressure), then the output voltage of the divider circuit will change in response.

For example, we can create a dimmer (for a light) or a volume control (for a speaker) by making one resistor into a variable resistor that the user controls. As the resistance changes, the voltage to the light or speaker changes as well.

Or, conversely, we can use a variable resistor that reacts to an external stimulus and then monitor the output voltage of the divider. Suppose, for example, that we had an object whose resistance changed as the light intensity changed; as the voltage output increased or decreased, we could use these values as a sort of light intensity meter. Or, suppose we had a resistor that responded to pressure; in this way, we could create a weight scale.

Return to the fixed voltage divider circuit diagram, Fig. 8. Replace resistor $R_2$ with one of the photoresistors.

A CdS based photoresistor. You may have seen these in night lights.

• As you cover/uncover the resistor or as you shine light directly on it (either from the provided lamp or from the flashlight feature on your cell phone), how does your output voltage respond? (Be as detailed as you can be!)
• Is your circuit sensitive enough to tell when a light is on or off? Could you use as a light intensity meter?

Next, replace $R_2$ in your circuit with the piece of Velostat (a pressure-sensitive material).

• How does the output voltage respond as you push on the material? (Be as detailed as you can be!)

Using the provided masses, make measurements of your output voltage as a function of mass placed onto the pad.

• How could this device be used to make a weight scale?

Creating a dimmer switch

Now that you've seen how a fixed voltage divider and an adjustable voltage divider work, it isn't too large a leap to translate these concepts into a practical circuit like a dimmer knob (for controlling the brightness of a light) or a volume knob (for controlling the loudness of a speaker).

In such a case, all we need to do is replace one leg of the voltage divider with our device (the lamp or the speaker) and replace the other leg with a variable resistor.

An example of a common type of variable resistor – called a rotary potentiometer – is shown in Fig. 10. This device works like the slide wire above, but is rolled up into a more compact form. The total resistance between the first and third legs is fixed, but as you turn the knob, the resistance between the first and second legs (or between second and third legs) changes. In this way you can select any resistance between zero and the maximum value.

Figure 10: A potentiometer. The total resistance between the first and third legs (Green & Orange) is fixed, but as you adjust the knob, the resistance between the first and second legs (or second and third legs) changes (Purple to Orange or Green to Purple)

Turn the fixed voltage divider circuit above (Fig. 8) into a lamp dimmer circuit and chose an input voltage of $V_{in} = 50\textrm{ V}$.

• Use a 1k$\Omega$ potentiometer (connected between first and second legs) in place of $R_1$.
• Use the light bulb in place of $R_2$.

How does the lamp behave as you adjust the potentiometer? Is it brightest when the potentiometer is close to zero ohms or close to 1k$\Omega$?

Using the tools at your disposal – voltage measurements, current measurements, direct resistance measurements, your voltage versus current plot for a lamp from the start of the lab, and Eq. (2) above – determine the “effective resistance” $R_2$ of the lamp at the following points:

• When $V_{out}$ is at a minimum value.
• When $V_{out} = (1/2)V_{in}$.
• When $V_{out}$ is at a maximum value. (HINT: at this point Eq. (2) is no longer useful. Even for a non-ohmic device, we can still say that for a given value of voltage and current, the “effective resistance” at that point is $R = V/I$.)

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!

Answer the questions/prompts below in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write the answers to these questions by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

The conclusion is your interpretation and discussion of your data.

• What do your data tell you?
• How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
• Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
• Do your results lead to new questions?
• Can you think of other ways to extend or improve the experiment?

In about one or two paragraphs, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.