1. Electric Circuits

Everyone needs some level of understanding of such devices - you don't want to pay an electrician to fix your lava light, you don't want to damage your eyes by incorrectly "jumping" your car battery, and someone's gotta get the grid back up after the storm ends.

We'll study circuits thoroughly enough to deal with the first two situations. And we'll get a good start on preparing those of you who might want to fix power systems, or design them.

1a. A Simple Circuit

Paper to write down your observations
2 fresh size D battery
1 double battery holder (Yours will differ)
wires with alligator clips
bulbs and bulb holders
1 500mA ammeter
1 digital voltmeter
30 gauge nichrome wire
1 1-amp diode (Yours will differ)
1 convex lens
1 momentary switch (Yours will differ)
1 pencil lead (mechanical pencil)

Note: Your battery holder will look different from the one shown here and in the images that follow. The use of the new holders will be explained in class. My appologies for not updating the images in The PC.

A. What makes a circuit?
Here's a bulb, a battery and a wire shown in picture form. These three objects and anything else you might put in a circuit are called circuit elements.

1. We want to make the bulb glow using one bulb, one wire and one battery. To do this you need to make contact between various parts of these three objects.

Look at the bulb. How many different parts are there to touch? There is the glass globe, the metal threads, a glass ring (shown green), and a metal tip. Draw a picture of the bulb as in Figure A. Draw short lines pointing to each of these four parts.

Repeat this process for the wire and the battery. Each of these two has three different parts. You've now identified all the parts that could be brought into contact.

Using trial and error, make various connections among these 10 points until you get the bulb to glow. When you find a combination that works, draw a picture showing the three circuit elements, and indicating how they are connected. Here's an example. You may use it as a guide. (I used tape at the bottom.) Here's a sketch. The tip of the bulb is touching the + end of the battery. The ends of the wire are touching the threads of the bulb and the negative end of the battery.

Find three other arrangements that will also make the bulb glow. You'll find this to be harder than expected. Just remember, if you change the connections in any way, it's a different arrangement. (Well, we won't consider turning the wire around to be a new case.) Draw pictures of each of these.

2. Along the way you found many arrangements that did not light the bulb. Draw pictures of two of these.

3. Look at your diagrams. Notice ways in which the bulb-lighting arrangements are similar and ways in which they differ from the arrangement where the bulbs did not light. In your own words describe what is necessary in order to get the bulb to light.

4. Holding wires to the bulb is a bit inconvenient. That's what the bulb holders are for. Screw your bulb into one of these holders. (This can be difficult.) Notice how the two metal parts of the bulb connect with the two clips on the bulb holder. From now on you can just clip your wires to these points.

5. Now check out the battery holder. Insert the battery in it and note how the ends of the battery connect with the little connecting tabs on the outside of the battery holder. If it has little metal tabs sticking out, bend them down so they point straight out. This is where you'll make connections to the battery.

B. Circuit Diagrams
Tired of drawing bulbs and batteries? There's a better way. We use an abstract system of simplified symbols to represent the various circuit elements. There are dozens of these symbols, but we'll just need to learn a few. Here are the ones we'll need. Note that we don't show the wire's insulation.

Here's one of the circuits you arranged earlier, represented as a picture and using a sketch (top) and circuit diagram (bottom). We'll stick with circuit diagrams from now on, they're much simpler. Make sure you see how both figures show the same thing.

Complaint Department!

You might say that you never set up this arrangement as you had only one wire. We'll assume that wires provide a connection between the elements they connect just as if you made the connection directly. A wire drawn in a circuit just indicates some sort of connection.

But your major complaint is that this is what it really looked like. The hardest part of all this is to think about these nice reasonable, well-behaved connections while using unruly wires and light-weight elements that go all over the place. Sigh.

Another point you might make is that the geometrical arrangement in these two diagrams is different. That is, the battery is on the left in one figure and on the top in the other. This doesn't matter. The only thing that matters is what connects to what. These diagrams all show the same circuit, electrically.

Circuit diagrams indicate electrical connections, not geometrical arrangements.

So far...

2. Electric Current

Take one end of a 10-cm section of #30 nichrome wire and hold it against the + end of your battery. Feel the wire at different points. You should not notice anything special.

Now, hold the other end against the - end of the battery (while still holding the first end in place). Feel the wire. Notice that it gets warm. Feel it at different points along its length. Now remove the wire from the battery.

Something seems to be happening in the wire and it seems to be the same all along the wire. Energy is being emitted as heat, so there must be some source of energy in our system and something is distributing it along the wire.

If we wait long enough (don't, it will quickly run down the battery) we know that the effect will diminish. We all know that the light of a flashlight will eventually dim.

We also found that we need a complete loop to make the bulb glow. Connections need to be made from the battery to the bulb and then from the other side of the bulb to the other side of the battery. It's as if something is going around a loop.

We could do a lot of speculating here, but we just don't have the tools to investigate. We'll adopt a model for electric current that we'll use as a tool as long as it helps us make correct predictions about our circuits.

Electric Current (preliminary) Electric current is a flow of something around a closed loop or circuit. The more current that flows, the brighter the bulbs in our circuits.

Current Direction

1. Using one bulb, one battery and two wires, set up a circuit that will make the bulb glow. Now reverse the connections at the battery. That is, disconnect the one attached to the + side of the battery and attach it to the - side. Do the reverse for the other wire. Do you notice anything different? Does the bulb still glow? How do you interpret what you observe?

2. Disconnect the wire attached to the - side of the battery and attach a third wire in its place. You should have two free ends. Here's a photo. Attach one free end to one end of the diode. Attach the other free end to the other end of the diode. What do you observe about the bulb? Now swap the wires connected to the battery as you did before. Any change? How do you interpret this result?

So which way does the current flow? Well... You're not going to like this. We have two answers.

The chemical forces acting in the battery result in one end of the battery having a positive charge and the other a negative charge. Or one end may be positive and the other more positive, or the same with negatives. Anyway, the two ends are at different potentials. We'll stay with the simpler negative and positive ends version.

These two ends set up an electric field in the space around the battery. When wires are attached, the field pushes electrons through the wires. This is called a flow of electrons.

Flow of Electrons Electrons flowing from a low potential to a high potential is called a flow of electrons. In our circuit, the electrons exit the negative pole of the battery, flow around to the positive pole and then through the battery to the negative pole where they repeat their journey. The electrons involved are the electrons that make up the battery and the wires. No electrons are produced by the battery.

That wasn't so bad. But there's more. Just as in our study of electrostatics, we reference everything to positive charges when we describe current.

For most purposes, a flow of electrons from low to high potential is equivalent to a flow of positive charges from high to low potential. We'll use this as our definition of electric current.

(Conventional) Current Positive charges flowing from a high potential to a low potential is called a Conventional Current or more simply a current.

We see positive charges flowing in plasmas such as lightning, in car batteries, and through the cell walls in our bodies.

But in our light bulb circuits it's just electrons flowing. But we will still refer to it as a current and we'll always refer to positive charges flowing from high to low potential. In the last animation we would say that electrons flow clockwise, but the current flows counterclockwise.

So far...

3. Conductors & Insulators

1. Write down the name of each item and beside it indicate whether or not the bulb glowed brightly.

plastic
nickel coin
wood
penny
glass
nichrome wire (10 cm)
paper
dime

2. What attribute do each of the materials that conducted the current have in common?

We refer to the items that allow the bulb to glow as electrical conductors. They allow the current to pass through them easily. The other items that don't allow (much) current to flow are insulators.

Which category do you predict pencil lead will fit into? Test this by connecting the clips to the ends of a piece of pencil lead, the sort that goes in a mechanical pencil. Gradually move the clips closer together along the length of the lead. Do the same thing with a 10-cm piece of #30 nichrome wire.

3. Do you see a change? Which category do pencil lead and nichrome wire fit in, conductor or insulator?

4. Based on your observations so far, which category does air fit in?

Switches
Connect the two free ends of your test circuit to the two ends of your switch. As you open and close the switch the light goes out, and then turns on, respectively. All that the switch does is let you choose between air (an insulator) and metal (a conductor) in that part of the circuit.

We'll usually put switches in our circuits to allow us to prevent the current from flowing except when we want it to. This will save our batteries.

Here's our simple circuit with the switch in action.

Bulbs
In our first look at bulbs we saw this pictorial representation. We later found that the bulb had two different metal parts, the threads and the tip, separated by an insulator, that had to be connected to the battery in order for the bulb to light. We drew it like this. Copy down this circuit sketch.

5. Based on what you now know about circuits, add the missing internal parts that you know must be present.

Let's see how you did. In your lab kit you have a convex lens. This is the glass lens that is thicker in the middle than at the edges. We'll use it here as a magnifier. Use your lens to have a good, close look at the bulb. Make any necessary adjustments to your diagram.

It looks something like this. And here's how you might draw it. You should see two metal support wires and a little wire (actually a tiny spring) connecting them at the top? This is the filament. Notice how the support wires are kept apart by a glass glob (shown as orange). Also notice how these two wires go down into another piece of insulation (shown as green). Most importantly you should notice how the two support wires connect to the threads and tip of the bulb.

So far...

4. Measuring Current

So how do we quantify current? It's just like the flow of water. If water flows very fast through a hose, about a gallon might pass each point in the hose every second. We'd say that the current of water is one gallon/sec.

If the current entering the hose is 1 gps, what would be the current 10 feet further along the hose? Would this answer change if the hose got wider or narrower along the way? .. if it had a leak, .. if there was a junction before that point?

Analogous to gallons with fluids, we quantify amounts of charge in Coulombs. So electrical current is measured in Coulombs per second.

Current One ampere (amp) is the current at a point in a circuit if one Coulomb of charge flows past that point every second.
	I is current in Amps Q is charge in Coulombs t is time in seconds

Note the emphasized words "past a point." Your water bill reflects the amount of water that has passed from the main pipe into the pipe to your house. The greater the average flow rate, the larger your bill. But when you turn on the faucet, every drop of water between your house and the city water tank moves. Fortunately it's not the total volume moving that you pay for. Just the amount that passes into your house.

Likewise with current, if you had an amp of current flowing in a small loop and an amp flowing in a larger loop, there would be more charges flowing in the second loop, but the same number flowing past each point in each circuit. Current does not refer to the total number of charges moving, rather the number moving past each point per second.

An ammeter is a device used to measure this current. Ammeters generally use magnetic effects to detect current. As the current must flow through the ammeter, the circuit must be opened at some point and the ammeter inserted in the gap. The current then passes through the ammeter.

We want to know the current flowing in this circuit. Now let's add the ammeter.

Here's the circuit diagram.

IMPORTANT! Notice the polarities (+/-) of the battery and ammeter. The ammeter should be labeled with +/- signs and/or with red(+) and black(-) terminals. The wire from the negative side of the battery is connected directly to the the negative side of the ammeter. Likewise, the wire from the positive side of the battery is connected, after passing through the bulb and switch, to the positive side of the ammeter.

Failure to use this pattern will cause the meter needle to deflect in the wrong direction and damage the meter. Whenever you set up a circuit with a meter, check for the proper alignment and then briefly hit the switch to check that the needle moves to some point within its operating range. If it deflects in the wrong direction or goes all the way to the other side ("pegs"), check your wiring and repeat.

Meter Polarities The + and - sides of any meter must be connected to the + and - sides of the battery, respectively. These connections may be indirect, that is, other elements may be present within the paths back to the battery.

1. You try it. Set up the circuit we've been observing. (Don't close the switch yet.) Connect a wire from the + terminal of the battery to one side of the bulb. Connect another from the other side of the bulb to the switch. Connect another wire from the other side of the switch to the + terminal of the ammeter. Connect another wire from the - terminal of the battery to the - terminal of the ammeter.

2. Close the switch for a few seconds. You should get a deflection of the meter needle to about the middle of its range. (The needle will jump around due to the imperfect connections between components. We can't do much about this. Just learn to live with it.)

Note that the meter is calibrated in milliamps (mA). The  mA label on the meter indicates that a full-scale reading is 500 mA. That is, if the needle goes exactly to the last mark on the right, the current is 500 mA or .500 A.

3. What is the current through this ammeter in mA and amps?

4. Try this one.

Record the current in your circuit, in amps, below.

5. We'll add a second battery in series with the first one. Here's the circuit diagram. See if you can figure out how to connect the real thing. Remember, just tap the switch first to make sure you don't damage the meter if your connections are incorrect.

Here's what it should look like. In case you're puzzled with how the battery is hooked up, here's a close up of the batteries. Notice how the the + end of the top battery connects to the - end of the lower battery at the upper left. The - end of the upper battery connects to the ammeter. The + end of the lower battery connects to the bulb.

(Don't try to do more with this relationship than this at this point. Important properties of the bulb also change when we add the second battery. We'll deal with this later.)

Does Current Get Used Up

Earlier we observed that the bulb was emitting light; energy is leaving the system. Since current appears to be related to energy, maybe the current gets used up as it flows around a circuit.

In the preceding activity we just measured the current at one point. Maybe if we measured it at other points we could document this gradual loss of current.

In our two-battery circuit, there are three distinct points where we could insert the ammeter ­ between the - side of the right-hand battery and the bulb (as in the figure), between the + side of the left-hand battery and the bulb, and between the batteries. (The switch just acts like a wire when it's closed.) We've already tried the first of these arrangements ­ between the - side of the battery and the bulb.

7. Sketch the circuit diagrams (2) for the other two arrangements.

8. Set up each of these and record the ammeter readings below.

The current is the same at all points around a current loop with no branches.

We'll find that this is always true, but the plot thickens when we start to find branches in our circuits. We'll get to that later.

How about inside the battery, the bulb, etc? Although we can't measure it with our meters, the current is the same inside the battery and bulb as it is in the wires connecting them into the circuit.

So far...

5. Measuring Voltage

Electrical Potential If a Coulomb of charge at a point has one Joule of potential energy, that charge is at a potential of one volt.
	V is voltage in Volts PE is potential energy in Joules Q is charge in Coulombs

Aha! Maybe this is where we'll find our energy. Suppose we replaced the ammeter in the circuit we used before with a voltmeter, like this. Well, not quite. Don't even try it.

Here's the deal. The voltmeter is designed to measure the change in voltage, the potential difference or voltage drop, between a pair of different points in the circuit. You don't open up the circuit and let the current pass through it like with an ammeter. Instead you clip a wire from the + terminal of the voltmeter to one point in the circuit and a wire from the - terminal of the voltmeter to some other point closer to the - pole of the battery. The voltmeter will then tell you the voltage drop between these two points.

Let's try it.

1. Set up your basic bulb and battery circuit.

2. Now attach the voltmeter across the bulb.

Here's how to set up the voltmeter. We'll always plug in the red and black wires as shown when measuring voltage. Note that the knob pointer is pointed at the 2 in the DC voltage area. (The other voltage area with the ~ is AC voltage.) If you're measuring more than 2 volts (when you use 2 batteries in series) move the pointer to the 20.

The circuit should look something like this. (The two wires in my hands are coming from the voltmeter.) Note that the + side of the battery is on the left and attaches to the top clip of the bulb holder. The + red probe from the voltmeter is touched to this clip. It's hard to see, but the - black probe from the voltmeter is touched to the other side of the bulb.

3. Close the switch and see if you get similar results. If not, check your wiring and try again.

4. How does your voltage drop reading compare to the rated voltage of the battery, which should be about 1.5 V?

What our voltmeter is reading is the voltage drop, the drop in voltage (energy/charge) as the charges pass through the circuit. All along the circuit the charges lose energy, so there's a continuous drop in voltage.

The rated voltage of the battery is called the Electromotive Force, Emf of the battery. This is a measure of the increase in voltage (energy/charge) as the charges pass through the battery.

Emf and Voltage Drop The Emf (electromotive force) of a battery (or other voltage source) is the voltage increase (energy/charge) provided to the charges as they pass through the Emf source. The Voltage Drop is the loss in voltage (energy/charge) as the current passes through the circuit.

What would energy conservation suggest about the relationship between these two voltages? If the Emf is the voltage supplied by the battery and the voltage drop is the loss in voltage in the circuit, surely the Emf must be at least as large as the voltage drop.

In our one-bulb circuit, the voltage drop seems to (approximately) match the Emf (actually a bit less). What if we add another bulb? Suppose we leave our voltmeter attached across our bulb and then add a second bulb to the circuit. Here's the circuit diagram. What do you think will happen to the voltmeter reading?

5. Give it a try. Set it up and throw the switch.

6. Surprise? You should be able to make a couple of predictions.

1. What will the meter read if you attach it across the other bulb?
2. What will it read if you attach it across both bulbs?

7. Try it. Make both measurements.

8. Put your observations into words.

Kirchhoff's Loop Rule Around a complete circuit the sum of the voltage increases equals the sum of the voltage drops. For our circuit, this means that the Emf (an increase in voltage) equals the sum of the voltage drops across the bulbs.

Measuring the Battery's Emf
The voltmeter measures voltage drop. The Emf is a voltage increase. How could we use our voltmeter to measure Emf? Look back at Kirchhoff's Loop Rule and think about this diagram.

Consider the small loop formed by the battery and voltmeter. According to Kirchhoff's Loop Rule there must be a voltage drop in the voltmeter equal to the Emf of the battery. So whatever the voltmeter reads is equal to the Emf of the battery. Note that it is also equal to the voltage drop across the bulb.

Voltage Drops in Wires
In our investigation above we basically ignored the wires. Would there be voltage drops in them too?

9. Check and see. Try measuring voltage drops across some of the wires. Something like this, for example. Here's the movie. What do you find?

Kirchhoff and Energy Conservation
Kirchhoff's Loop Rule states that the voltage drops in the circuit equal the voltage increases. The voltage changes are changes in energy per charge. So as a charge goes around a circuit it is given energy by the batteries, only to have it taken away by the "load".

What if there were no load? Our wire didn't seem to drop the voltage any. What if we just connected the wire from pole to pole of the battery? You did that earlier & found that the wire got quite warm. In this case there was a full 1.5-V drop in the wire. In our bulb circuit there was almost no drop in the wires. Keep this puzzle in mind as we proceed. We will resolve this issue.

So somehow, whatever is in the circuit will be able to dissipate the energy provided by the battery. In uncontrolled situations, such as short circuits, this can lead to disaster. We will of course demonstrate this in class.

So far...

6. Power - A First Look

Power Power is the rate at which work is done or energy is transformed or transferred.
	P is Power is in Watts W is Work (or Energy) in Joules t is time in seconds

As charge flows around the circuit, it gains energy from the Emf source and loses all this energy to the load. The power of the battery is the rate at which it provides energy. The power of the load is the rate at which it dissipates this energy. Let's create a more electrical version of the power equation.

Electrical Power
	P is Power is in Watts I is Current in Amps V is voltage in Volts

Let's see how this works with our bulbs. Set up this circuit and measure the current and voltage for bulb a. Record these below (I₁ and V_a1) and calculate the power (P_a1) of bulb a.

Make a mental note of the brightness of bulb a.

Add bulb b and notice how dramatically bulb a dimmed. Obviously each bulb is much dimmer than the single bulb was before. You may even have to dim the room lights to see any glow. How much dimmer? Well, your eyes can't tell you, they're non-linear, but the math can.

Measure and record the current (I₂) and two voltages (V_a2 and V_b2). Calculate the power of each bulb (P_a2 and P_b2).

I(A)	V(V)	P(W)
I1=	Va1=	Pa1=
I2=	Va2=	Pa2=
	Vb2=	Pb2=

Can you see from the table why bulb a dimmed so much? Both the current through it and the voltage across it decreased.

1. If the bulbs were identical you'd find that both the current and the voltage were cut in half. For this case, what would be the value of the quantity P_a2/P_a1? That is, by what factor is the power decreased?

So if the power of bulb a is quartered and both bulbs have the same current and voltage drop, the total power of the two bulbs together is half the power of the single bulb when it was alone.

Why do the current and voltage change? To understand this new observation we need to study a new aspect of electrical current, electrical resistance.

So far...

7. Electrical Resistance

7a. Current vs. Voltage

Here's an animated view of the apparatus.

1. A 1.1-m (approx.) length of nichrome wire is taped to the table beside a meter stick. The wire from the negative side of the 1.5-V battery connects to the ammeter. Another wire connects the ammeter to the right end of the nichrome wire, at the 100-cm mark on the meter stick.

2. The wire from the positive side of the battery is connected to a switch. A wire from the other side of the switch connects to the nichrome wire at the 0-cm mark.

3. A voltmeter is clipped to the 80 and 100-cm points on the nichrome wire. Note that the ammeter wire clip will hold the negative voltmeter probe in place. The positive voltmeter probe can be held with the clip on an extra wire. Look back at the details image to see this better.

(Note: Alert diagram watchers may be confused by our connecting the - of the voltmeter to the + of the ammeter. The rule about these connections refers to the battery not any previous circuit elements. That is, the - side of each meter is connected to a wire coming from the - side of the battery. What happens in between is irrelevant. But thanks for caring.)

... Step through the frames to see how you will be moving the clip attached to the left end of the nichrome wire. Here's a movie. Each time you move it you'll record the meter readings. Not yet.

4. Set up the apparatus in the initial configuration (Frame 1 of the animation).

5. Briefly close the switch and record the current and voltage readings in the first row of the table below.

So what's going to happen when you move the left clip? As you move the clip, the voltage drop across the 20-cm test segment and the current through it are going to change. We'll learn why later, but you saw this happen before when we investigated power. We're just using this technique to see how the current through a conductor is related to the voltage across it.

Changing the length of the left end of the wire is just our way of bringing about change in the right end, the test segment, which is where we'll concentrate our attention. Just note that your voltage readings are for the test segment. The current readings are, of course, the same for any point in the circuit, but we'll think of them as the current in the test segment.

6. Move the clip to the next position in the table and read your meters. Repeat to get the data for the rest of the table.

+ Clip Position (cm)	Vtest(V)	Itest(A)
0
10
20
30
40
50
60
70
80

7. From your data table you should be able to see that both the voltage and current are rising together. Use GA to plot Current vs. Voltage to get the exact relationship.

8. What is the relationship between the current flowing through our test segment and the voltage drop across it?

Current vs. Voltage Drop The amount of current passing through a conductor is directly proportional to the voltage drop across it and vice versa.

This explains why our bulbs got brighter when we added more batteries. More Emf means more voltage drop which means more current which means more power.

9. When a 9-V battery provides the Emf for a certain circuit a current of 2.4 A flows. What current would flow if the battery were replaced with a 1.5-V battery?

7b. Current vs. Resistance

Notice that this time we've dropped the voltmeter. Now we're looking at the length of the wire through which the current flows. ... You should see that as we move the alligator clip, the current just flows through the part of the wire between the clips. This length, L, decreases as we move the left clip.

We want to look at current vs. the length, L, of nichrome wire through which the current is flowing. You don't need any new data. Just copy your current readings into this new table. You should be able to figure out the length values.

+ Clip Position (cm)	Length (cm)	I(A)
0
10
20
30
40
50
60
70
80

This time the data go in opposite directions. The longer the wire the current flows through, the less the current. This is much the same thing that you'd find with water flowing through a hose. The longer the hose, the less the current. You can fill a bucket up faster at the faucet than at the end of a long hose. Try it.

Here's the plot I got of my data. This is a good example of an inverse proportion. The graph isn't exactly right since we should include the effect of the rest of the wiring, but the relationship seems clear enough.

For a water hose, the resistance to the flow is due to the frictional drag of the water against the side of the hose. The longer the hose, the more the resistance. In the case of electrical current, it's more complex, but the result is similar. We call this resistance to the flow of current, well, err, resistance. The electrical resistance is directly proportional to the length of the conductor. Also, a good conductor has low resistance.

Current vs. Resistance The amount of current passing through a conductor is inversely proportional to resistance in the conductor.

Combining our two observations, that the current is directly proportional to the voltage drop and inversely proportional to the resistance gives us Ohm's Law. The units of resistance are chosen such that we can write this law as an equation with a constant of proportionality of unity.

Ohm's Law The current through a conductor is directly proportional to the voltage drop across it and inversely proportional to its resistance.
	I is current in Amps(A) V is voltage drop in Volts (V) R is resistance in Ohms (Ω)

This equation describes the interrelationship among I, V, and R in a circuit. As you have no experience with this phenomenon, the equation probably has no meaning to you. In words, it says

"When a current flows through a section of wire, the voltage will drop by an amount I x R. The more current flowing, the more the voltage drop. The greater the resistance, the greater the voltage drop. Or, the current is increased by increasing the voltage drop or decreasing the resistance."

So just what was the resistance of our nichrome wire? It depends on the length. Let's look at our 20-cm section. Here's our graph. Can you find the resistance of the 20-cm section using this graph and Ohm's Law?

Of course when we're not in the lab we tend to figure things out with equations that we get from the lab. Try this animated activity. I got the values I refer to with a thicker section of nichrome wire so the numbers are quite different. WARNING: There's a lot to learn from this one. You might go through it more than once.

Got it? We have other questions to answer about this simple relationship. We'll save them for later.

7c. Power Revisited

Now that we have Ohm's Law we can expand a bit on our original power equation.

Power Equations P = IV P = I2R P = V2/R

So far...

8. Series Circuits

You should expect your intuition to be helpful for a while, but a crash is inevitable. It is crucial that you are always actively working out a model for understanding these circuits. There are rules, but they can be subtle.

As we add more components we want to concentrate on three key quantities: current, voltage drop, and resistance. We'll also consider power since it's a good indicator of current and voltage drop.

Whenever we can, we'll use bulbs to provide resistance, as their differences in brightness are instructive. Unfortunately, bulbs are not well behaved; their resistances vary a lot with temperature. So we'll often use sections of nichrome wire since their resistance is fairly constant for the currents we're using.

Most importantly, we know that for such a wire the resistance is directly proportional to its length. So if L₂ = 2 L₁, then R₂ = 2 R₁

We'll look first at Resistances in Series.

Elements in Series Circuit elements connected end to end, with no branching between, are said to be connected in series. Below we represent three resistances in series. They might be bulbs or sections of wire, for example.

8a. Current in a Series Circuit

Consider these circuits containing similar resistors.

We have two questions to investigate:

• How does the current, I_a, through the battery in the first circuit compare to that in the second circuit, I_b?
• How does the current I₁, through R₁ compare to the current I₂, through R₂? And how do these currents compare to I_b?

4. Set up both of these circuits using bulbs. Don't use meters. You can get a good idea about the current from the brightnesses of the bulbs.

A note about equipment: The construction of the sort of components we're using is never very consistent. No two "identical" bulbs are likely to be made the same. So don't be picky about small differences. For example, the brightnesses of two different bulbs may not look exactly the same, but if they are close we won't quibble. We'll say they have the same brightness. When they are different, it will be dramatic.

5. Make the observations you need to test questions 1-3 above.

6. Put your findings into words. This is an important step. You can't memorize what you're learning. You're learning how to think about circuits and apply a few rules to make predictions.

Here's the first of three important observations about series circuits.

8b. Voltage in a Series Circuit

So, what do you think? Both circuits have 1.5-V batteries. Think about what you found earlier about changes in current when resistors were added in series. Do you expect similar results here with voltage? Let's make some predictions.

Fill in the blanks with <, >, =, or + for each question below.

3. Let's get some measurements. Cut two lengths of nichrome wire - one 22-cm long, the other 42-cm long. Set up Circuit 1 using just the 22-cm wire. Adjust the alligator clips on the hook-up wires so that there is 20 cm of wire between them. The extra 2-cm gives you a centimeter to clip to at each end. You may want to use some tape to hold things in place.

Apparatus Note: Since we'll be using these sections of wire a lot let's use some shorthand. When you're asked to use, say, a 30-cm wire, you'll make it 2-cm longer for clip-on points, but we'll still call it a 30-cm wire since the current will just flow through the 30 cm.

Also, we'll simplify our drawings now by omitting the switches. You still need to use one in all circuits. We'll just leave them out of the drawings.

4. Measure V_ab for this circuit. (Touch the + voltmeter probe to point a and the - probe to b.) Record your reading below as V_ab1.

5. Now set up Circuit 2 using the two nichrome wires as the two resistances. Join them together with a hook-up wire such that the current flows through a 20-cm section of wire 1 and 40-cm of wire 2.

6. Measure V_ab in this circuit. Also measure V₁ and V₂.

How could Kirchhoff's Loop Rule help us understand this result? This rule says that the sum of the voltage drops equals the total Emf. Since we have an Emf of about 1.5 V in each circuit, the voltage must drop by that amount in each circuit.

Another important way of looking at this is to say that all the energy given to the charges by the Emf source is dissipated as the charges make their way around the circuit. No energy is ever left over.

8. How about question #2 - the relationship between V_ab, V₁, and V₂? Put your findings into words. Here's the circuit again.

Here's why. With current we were measuring a flow of something that traveled through a succession of components. Conservation of charge insists that the flow rate be the same at all points. With voltage we start by giving each charge some energy (Emf=Energy/Charge). As these charges travel around the circuit they dissipate their energy a little at a time. So each component has its own voltage drop. They all add up to a total drop equal to the Emf.

What if we added a third wire in series? We've already used up our 1.5-V Emf. Something will have to give. Let's try it.

9. Cut a 60-cm nichrome (62-cm) wire and add it to your circuit after the 40-cm wire. Predictions first. How will the voltage drops be ranked? Type in V1, V2, and V3 for the 20-cm, 40-cm, and 60-cm wires respectively to show your predictions.

But here's something new. The original voltages, V₁ & V₂ decreased. They had to, to "make room" for the new wire. Likewise, in our study of current, the bulbs got dimmer when we added more in series - the current decreased. So voltage drop and current aren't properties of the wires. So just how do the wires "know" how much to drop the voltage?

To predict the specific voltages and currents we need to investigate the one measurement that is a property of the wires, the resistance.

But first, here's an important question about lingo. Consider the following statement.

"I hooked up a 10-ohm resistor to a 1.5-V battery and got a current of .15 A. When I added a second 10-ohm resistor in series, the current dropped."

11. Draw these two circuits, indicating all currents and voltage drops. Clearly discuss the use of the word "dropped" in the quote above. Compare the use of the term here with the term "voltage drop."

12. Test your skills so far on this problem.

8c. Resistance in a Series Circuit

We'll use our nichrome wires for resistors. The first thing we need to do is find their resistances. We can use Ohm's law if we know the current through them when there's a given voltage drop across them. We'll set up a circuit to measure these values.

1. Hook up a simple circuit with a battery, switch (not shown), ammeter, and 20-cm of nichrome wire. (Use the 22-cm piece as before.) Attach a voltmeter across the nichrome wire.

2. Record your voltage and current readings below. Calculate the resistance of the 20-cm wire.

3. Repeat 1 & 2 for the 40-cm wire.

Voltage, V	Current, A	Resistance, Ω
V20 =	I20 =	R20 =
V40 =	I40 =	R40 =

4. Do your numbers make sense? You should have found the 40-cm wire to have about twice the resistance of the 20-cm wire.

OK, you do the physics. Predict what will happen when you connect the two wires in series. (Don't hook it up yet.) R_ab is the total resistance between points a and b.

- Ross Perot

So the logic goes like this:

Vab = 1.5 V	Same as Emf
Rab = 15  Ω	R1 + R2
I = Vab/Rab = .1 A	Ohm's Law

That's our third observation about series circuits. Here's the list using the term R_eq which stands for equivalent resistance. This is the resistance that could replace the original resistors with no effect on the rest of the circuit.

Resistance in Series

Variable Resistance

9. Set up a series circuit using a battery, a switch, a bulb and the 40-cm nichrome wire.

Close the switch and note the brightness of the bulb.

10. How could you change the brightness of the bulb by moving just one of your clips?

Try it. Note the effect on the brightness of the bulb.

11. Describe the reason for the change in terms of resistance, current, voltage, and power. Take some time with this. It's not obvious.

12. As you move the clip, R_eq

increases

decreases

is unchanged

13. As you move the clip, the voltage drop across the load

increases

decreases

is unchanged

14. One more time. Describe the reason for the increased brightness in terms of resistance, current, voltage, and power.

1. V_ab remains const. (equal to Emf)
2. R_wire decreases. (due to decreasing length)
3. R_eq decreases. (from R_wire+R_bulb to R_bulb)
4. I = V/R increases. (from V/R_wire+bulb to V/R_bulb)
5. P_{load & battery} = IV increases (due to increasing I)
   

The sliders and knobs on this audio board are all potentiometers or "pots". They vary the volume of the sounds being controlled by the board by changing the resistance and hence the current being sent to the speakers or tape machine. One last thing. We found that moving the clip increased the circuit current. We also noted that the voltage drop across the circuit remained equal to the Emf. But what about the voltage drop across the bulb. This is the one that's key to the power dissipated by the bulb.

Try it yourself. Set up this circuit and use your meters (not shown, but you know how) to fill in the following table for the three situations in the animation.

	I(A)	Vab(V)	Vbc(V)	Pbulb(W)
1
2
3

You should clearly see from your results that there are two reasons for the increase in brightness of the bulb. The current is increasing, and the bulb is getting an increasingly larger share of the voltage.

V_ac is constant and V_ab is decreasing, so V_bc must increase by equal amounts.

So we could say that the rheostat varies the current in the circuit by changing the total resistance and changes the voltage drop across other parts of the circuit by changing the voltage drop across itself.

Temperature Effects on Resistance

	Rbulb(Ω)
1
2
3

Clearly, the resistance of the bulb is increasing. But why? This is a temperature effect. Remember how Thomas Edison worked so hard to find just the right material for a filament? He needed something that would get very hot and glow but not burn up, melt, or evaporate away quickly. As the bulb heats up, the large increase in molecular motion interferes with the motion of the electrons. Our wires are carrying the same current as the filament; they just don't heat up as much for various reasons. More later.

Let's add this to the other factor that we've found to effect the resistance of a conductor.

Resistance of a Conductor The resistance of a conductor is directly proportional to its length. For most conductors the resistance increases with temperature.

On the dashboard of your car you may have a gauge that indicates the temperature of the engine. There are other such gauges that control various functions of your engine. Many of these gauges use thermistors, resistors that change in resistance with temperature. As the car's engine heats up, the current passing through a thermistor decreases. The amount of this current is used by your car's computers to fine-tune the engine's operation. Your dashboard meters are just ammeters that indicate the current flowing through the thermistor.

Example Questions

Here are the facts. You need to use them to construct a strategy to find an unknown. You'll often use all these relationships.

15. Find the values requested in each step in this circuit. Give it your best shot before you have the animation show the solution. (V₁ is the voltage drop across R₁, etc.)

Now that you've seen the solution, you should notice a few things about the method. We drew a new diagram with just the equivalent resistance. This diagram is just as valid as the original one. That is, the current and V_ab are the same in both circuits. That's why we use the term equivalent resistance. As far as the rest of the circuit is concerned you could have a 2Ω and a 6Ω or a single 8Ω resistor between a and b.

In step 2 we used the new diagram since we knew that the 4-V drop was from a to b, so we needed the total resistance between a and b.

In steps 3 and 4 we had to return to the first diagram since it has the two individual resistors. We used ohm's law for each resistor individually.

You might think of using these different equivalent circuits just like equivalent equations. The equation

(2x + 6) (x - 4) = 10

is equivalent to

x² - x = 17

Anything you might do with either equation will produce the same result.

16. Try this one. Be sure to give it your best shot before you play the animated answer. Puzzling huh? You can't draw a second circuit since you have an unknown resistance. How can you find a resistance? Look back at the 5 equations we saw earlier. Which ones have R's in them? You need to work out a strategy and it won't be just one step.

17. A simple circuit is constructed using a bulb, a battery, and two wires. To get the most light, most of the resistance should be in the

wires.

bulb.

18. Why? Don't look at the answer until you've drawn some figures and had a good discussion. You might want to just assume some values and see where they lead you. You're looking for a way to maximize the power of the bulb.

So far...

9. Parallel Circuits

Puzzle over this one. Be sure and write down your predictions.

Most people have real trouble setting up circuits like this. It looks simple in the figure, but the figure is just an abstraction. You also don't know where to start - kind of like being given a model airplane kit with no step-by-step instructions. When you're doing wiring, remember what we learned earlier:

Circuit diagrams indicate electrical connections, not geometrical arrangements.

All the diagram says is that the current leaves the battery, divides after passing through bulb 1 and reconverges before returning to the battery. To accomplish this you just work your way step by step, making one connection at a time. Note that there is more than one way to make the connections. I'll always try to make the simplest arrangements.

Here's a movie that shows, step by step, how to put the circuit together. Don't play this as a movie (although it's fun to watch.) Instead read each lettered statement below, then step  to the corresponding frame of the movie. We'll simplify the directions by calling the side of a bulb closest to the + side of the batteries the + side of the bulb. The other side is the - side of the bulb.

Note that we have two batteries in series. The top of the left battery is +, the top of the right battery is -. The bottoms are connected with a wire.

1. Wire 1: + side of battery to + side of bulb.
2. Wire 2: - side of bulb 1 to + side of bulb 2.
3. Wire 3: - side of bulb 2 to - side of battery.
   (Bulbs 1 and 2 should light now.)
4. Wire 4: - side of bulb 1 to + side of open switch.
5. Wire 5: - side of switch to + side of bulb 3.
6. Wire 6: - side of bulb 3 to - side of battery.

Remind yourself of the question. Review your predictions. Drum roll please. Close the switch!

Guessed wrong didn't you? Most people don't get close on this one. Here's an animated view that's easier to see. Actually bulbs 2 & 3 pretty well go out. Weird.

How are you supposed to think about such a circuit? We need to study arrangements like bulbs 2 and 3 first, then we'll come back to this one.

Bulbs 2 and 3 are in parallel with each other.

Elements in Parallel Circuit elements connected with common end points are said to be connected in parallel.
Here we represent three resistances in parallel. When the current comes to such an arrangement, it divides. Some of the current goes through each branch.

Just as with series circuits we want to investigate three quantities: voltage drop, current, and resistance. We'll start with voltage.

9a. Voltage in a Parallel Circuit

Both circuits have 1.5-V batteries. Let's make some predictions.

Fill in the blanks with <, >, =, or + for each question below.

3. Let's try it. We want to set up circuit 2 in the figure using our 20 and 40-cm nichrome wires for R₁ and R₂, respectively. Set it up. Here's a photo. (Be sure to include the switch.) Notice that I've been careful to keep the wires from touching each other or coiling up. Sometimes you'll want to use tape to keep things in place.

4. Measure V_ab, V₁, and V₂. Record them below.

The key point here is that the voltage drop must be the same across both resistors since they start and end at the same point. This gives us the first of three important observations about parallel circuits.

Do you think this would change if we added a third wire in parallel? Nope, still the same. But there will be other things that change.

9b. Current in a Parallel Circuit

• How does the current, I_a, through the battery in the first circuit compare to that in the second circuit, I_b?
• How does the current I₁, through R₁ compare to the current I₂, through R₂? And how do these currents compare to I_b?

4. Set up this variation of circuit 2 using bulbs.

5. What do you think will happen when you close the switch, effectively changing from circuit 1 to circuit 2? Do you see that the switch adds in the second parallel section. So the battery current is I_a with the switch open and I_b with the switch closed. So what will happen?

Warning. You'll notice from the equal brightnesses of the bulbs that the current is the same in each bulb. But this is not true in general as we'll see shortly. If we used two different types of bulbs, the brightnesses would be different.

What we're after here is a comparison of the currents in circuits 1 and 2. Think about it. Before closing the switch, a certain current flows through bulb 2. After closing the switch the same current still flows through bulb 2 since it's still connected in the same way to the battery. But there is also an independent current (equal in this case) flowing through bulb 1. What do you think this will do to the current through the battery?

8. Set up this circuit by replacing the bulbs in your circuit with wires and adding an ammeter. Use a 20-cm nichrome wire for R₁ and a 40-cm nichrome wire for R₂.

9. Make the observations you need to test questions 1-3 above. They are repeated again below for your convenience.

Before you proceed, be sure that you understand that closing the switch adds resistance R₁ to the circuit and changes the battery current from I_a to I_b.

You'll need to measure the four currents listed below. I_a and I_b shouldn't be too hard. Just put the meter where it's shown in the diagram and measure with the switch open, then closed. Be sure to run the wire from the + side of the battery to the + side of the meter.

Measuring I₁ and I₂ are a little trickier. These currents only exist in their separate branches. If you disconnect the clip attached to the + side of a resistance, the ammeter can go in the gap. You'll need to add an extra wire. Here's a movie of the setup for measuring I₁.

Ia = A	Ib = A
I1 = A	I2 = A

Here's the second of three important observations about parallel circuits.

11. Let's try an example problem to see how these first two observations work for us. Be sure to try to work out each question before you look at the solution.

12. Let's try another example problem with a little different mix.

What you've just discovered is summarized by the following:

Kirchhoff's Point Rule The sum of the currents flowing into a point is equal to the sum of the currents flowing out of the point.

9c. Resistance in a Parallel Circuit

So what about that last statement, about the equivalent resistance of resistances in parallel? That is, if we have two resistors in parallel, what single resistor could replace it and still have the same battery current. What do you think the equation will look like? We want an equation that relates R_eq, R₁, and R₂. Let's look back at the problem we just finished. Here's what the final result looked like.

We want to find the equivalent resistance of this circuit. Remember, R_eq is the single resistance that could replace R₁, and R₂, with no change in the rest of the circuit. Specifically, I must remain equal to 6A.

1. Let's work through the logic of this circuit.

So what do you think. Give it a shot. Monkey around with the three resistance values a bit. There is an equation hiding here, but it's a bit odd looking. Here are the first two equations. The one you want has all three R's, a + and an =. Come on. Figure it out using our numbers, then write it for the general case with just R_eq, R₁, and R₂.

1) V_ab = V₁ = V₂

2) I = I₁ + I₂

So here's the final result that tells us how to simplify series and parallel circuits. (We'll see why we want to soon.)

Here's what the chart tells us.

Series Resistance

1. The voltage drops in steps as current flows through a series of of resistances. The total voltage drop equals the sum of the individual drops.
2. The current is the same through all resistances in series.
3. The total or equivalent resistance of a series of resistances equals the sum the individual resistances.

Parallel Resistance

1. The voltage drop is the same across all all resistances in parallel.
2. The current entering a group of resistances in parallel divides according to the resistance of each branch. The current flowing in (and out) of the junction points is equal to the sum of the current in the individual branches.
3. The inverse of the equivalent resistance of a group of resistances in parallel equals the sum the inverses of the individual resistances.

So far...