Chapter 11: Practical Circuits

Welcome to Last Minute Lecture.

This free chapter overview is designed to help students review and understand key concepts.

These summaries supplement, not replace, the original textbook and may not be redistributed or resold.

For complete coverage, always consult the official text.

Welcome to this deep dive.

You know, our mission today is to really master Chapter 11 Practical Circuits straight out of the Cambridge International AS and A Level Physics Coursebook.

Right.

And we are going to be building a completely ground up understanding of how real world circuits behave.

Exactly.

From the power source itself, all the way down to like super precision measurement techniques.

We're going to treat this like a one -on -one tutoring session for you.

Yeah.

Unpacking not just how to calculate the results, but why the physics actually works the way it does.

But before we get to the formulas, I want to start with a historical mystery that is just, it's mind blowing.

Oh, I know where you're going with this.

The Baghdad Find.

Yes.

So in 1936, archaeologists are digging near Baghdad and they unearth this small pitch sealed clay pot.

Inside the pot is a copper cylinder and inside the cylinder is an iron rod.

Right.

And if you take that exact arrangement and fill the pot with a common acidic liquid, like say vinegar, it produces a potential difference of about 1 .5 volts.

Which is, you know, incredibly close to a standard modern AA battery.

Right.

And this raises a massive historical question because we usually give Alessandro Volta all the credit for inventing the battery in the year 1800.

Yeah.

With his alternating discs of copper and zinc and brine soaked felt.

Exactly.

Right after his rival Galwani made a dead frog's leg twitch.

But this Baghdad artifact suggests ancient technologists might have beaten Volta by over a thousand years.

Oh.

Possibly using this chemical battery to electroplate metal objects with gold.

What's fascinating here is that the physical principles of electricity don't change based on who is discovering them.

Very true.

Whether we are looking at a 2000 year old clay pot filled with vinegar or, you know, a cutting edge lithium ion cell in a modern electric vehicle, every single real world power source shares the exact same hidden physical flaw.

Okay, let's unpack this because this limitation really reframes how we think about circuits, doesn't it?

It completely does.

We usually imagine a battery is this perfect magical box that just effortlessly pumps out voltage to the rest of the circuit.

Right.

The ideal power supply.

But if the chemical reaction inside the battery is the thing generating the charge, well, those chemicals must also physically be in the way of the charge trying to flow out.

That is the core of it.

We call this internal resistance and it's represented by a lowercase r.

Internal resistance.

So think of like a shipping fee or an energy tax that the battery charges itself before the electricity even reaches the external circuit.

I like that analogy.

Charges moving around a circuit don't just magically teleport from the power supply to the wires.

They actually have to pass through the power supply itself.

Right.

So the chemical paste in a dry cell or the liquid acid in a car battery or even the internal wiring of a bench top power supply, all of these materials resist the flow of charge.

It's like trying to run a sprint through a crowded stadium concourse.

Yeah.

The battery gives the electrons the energy they need to run.

But the moment they start moving, they immediately have to shoulder their way through the battery's own chemical suit before they even reach the exit doors.

They are expending energy just to get out.

And we can actually measure the physical consequence of that struggle.

Right.

The text talks about a real world test for this, which I love.

Exactly.

As the charges gain electrical energy from the chemical reaction, they immediately lose some of that energy doing work against the internal resistance.

Right.

And that lost energy transforms into internal thermal energy.

Which means the battery gets hot.

Yeah.

Yeah.

If you take a standard cell, feel its temperature, and then use it to light a small torch ball for just 15 seconds, you can physically feel the temperature rise.

It's immediate.

That heat is the literal tactile evidence of charges colliding and doing work against the internal resistance.

So to handle this mathematically, we have to change how we model power sources in our circuit diagrams.

We can't just draw a simple battery symbol anymore?

No, we model a real power source as two separate things enclosed inside a dashed box.

Okay.

What are the two things?

First, a theoretically perfect cell that provides the total electromotive force, or EMF, that's represented by a capital E.

Capital E.

Got it.

And second, connected in series with it, is a small internal resistor.

And that is represented by a lowercase r.

So the EMF, that capital E, is the total theoretical energy per coulomb the battery could provide if there were no stadium crowd to push through.

Exactly.

But since the internal resistance, the lowercase r, is in series with the external circuit, what's called the outside wires and capital R.

Right.

Capital R for the external load.

If the total resistance is capital R plus lowercase r, how do we mathematically describe the current?

Well, following the standard rules of series circuits, the current, capital I, is the same everywhere.

Right.

So using Ohm's law, the total electromotive force equals the current multiplied by the total resistance of the whole system.

Okay.

So the equation is E equals I times the quantity in parentheses, capital R plus lowercase R.

Exactly.

E equals I times parenthesis R plus R close parenthesis.

And if we distribute that current across both resistors, we get E equals I times capital R plus I times lowercase r.

And that expanded equation tells a profound story about where the energy is actually going.

It really does.

Let's look at the two halves of that.

So the first part, I times capital R, is the current multiplied by the external resistance.

That is the terminal potential difference represented by V.

Eternal PD.

Yes.

It's the actual usable voltage you measure across the outside terminals of the battery.

Which means the second part of the equation, the I times lower case r, is the voltage that never made it out.

We literally call them the lost volt.

Lost volt.

Yeah.

So if we rearrange the equation, we get the fundamental rule of practical circuits.

V equals E minus IR.

Okay.

So the usable voltage, V, equals the theoretical maximum EMF minus the lost volts dropped across the internal resistance.

You've got it.

Let's visualize this with a worked example from the text.

Say we have a 6 .0 volt battery.

So that 6 .0 volts is our EMF, our capital E.

Right.

And we connect it to an external resistor of 13 .5 ohms.

When we do that, we measure a current of 0 .4 hour amps flowing through the circuit.

Okay.

So let's plug those numbers into our equation to find the internal resistance R.

So the equation is 6 .0 equals 0 .40 times 13 .5 plus 0 .40 times R.

Right.

And if you multiply 0 .40 by the external resistance of 13 .5, you get 5 .4 volts.

Oh, so 5 .4 volts is our terminal PD.

That's the V.

Exactly.

The battery is rated for 6 .0, but the external circuit only sees 5 .4 volts.

So do in the algebra.

6 .0 equals 5 .4 plus 1 .40 R subtracts by 0 .4 and we get 4 .6 equals 0 .40 times R.

And 0 .6 divided by 0 .40 gives us an internal resistance of 1 .5 ohms.

Perfect.

It's so cool how you can just deduce the invisible properties of the battery like that.

And the textbook brings in a really practical everyday scenario that drives people crazy.

A car battery on a cold morning.

Yes.

A typical car battery has an EMF of about 12 volts and an extremely low internal resistance around 0 .04 ohms.

Okay.

0 .04 ohms is tiny.

Right.

Under normal conditions, like just running the radio, the car draws a very small current.

So the lost volts current times 0 .04 is basically negligible.

The terminal potential difference stays very close to 12 volts.

But then you turn the key.

The starter motor engages and that motor requires a massive amount of power to physically crank the cold engine.

It suddenly draws around 100 amps of current from the battery.

Now apply the V

I R equation.

Okay.

The current I is 100 amps.

The internal resistance R is 0 .04 ohms.

Multiply them and your lost volts equal 4 volts.

That is huge.

The battery is dropping 4 volts.

Just trying to push that massive current through its own internal acid.

So your 12 volt battery is suddenly only delivering 8 volts to the entire rest of the car.

Which is exactly why if you try to start a car with the headlights already turned on, the headlights noticeably dim.

Right.

Because they are suddenly trying to operate on 8 volts instead of 12.

The voltage sags because the system is demanding more current than the chemical reaction can efficiently push through its own internal resistance.

Okay.

So how do we map this behavior out in a lab setting?

Because practical activity 12 .1 explains this and you can't just connect a digital voltmeter and call it a day, right?

No, because a high resistance digital voltmeter draws almost zero current.

So the lost volts are 0.

It basically just reads the pure EMF.

Exactly.

To find the hidden internal resistance, you have to force the battery to work.

So what's the setup?

You build a circuit with a power supply, an ammeter to measure the current, a voltmeter connected across the supply to measure the terminal PB,

and a variable resistor.

Ah, the variable resistor.

So by adjusting that, you force the battery to deliver different amounts of current.

Right.

And you record how much the terminal voltage sags each time.

And when you plot those results on a graph putting terminal voltage V on the Y axis and current I on the X axis, you get a straight line sloping downwards.

Because the physical behavior perfectly mirrors the mathematical equation for a straight line, Y equals MX plus C.

Okay, let's map that out.

In our case, the equation rearranges to V equals negative R times I plus E.

Oh, so the Y intercept where the current is zero, that gives you the theoretical EMF capital E.

Yes.

No current means no loss volts.

And the gradient, the slope of that downward line, is the negative internal resistance.

Negative R.

Exactly.

The steeper the slope, the worse the battery is at delivering high currents.

It is just a beautiful way to make the invisible properties of a battery entirely visible.

It really is.

But it highlights a major engineering hurdle.

Because internal resistance means a battery's output voltage is inherently unstable.

It changes depending on what you plug into it.

Right.

So if you are designing a sensitive piece of electronics that requires exactly 3 .0 volts to run,

you cannot just hook it up to a 6 .0 volt battery and hope for the best.

Definitely not.

You need a way to force the voltage to be exactly what you want, regardless of the battery fluctuations.

We need a way to slice the voltage down to size.

And the simplest, most elegant way to do this is with a potential divider.

Okay, describe the setup.

You take your power supply and connect it across two resistors in series.

So let's picture this.

We have an input voltage, say 10 volts.

We connect it to resistor one and then resistor two in a single loop.

Because they are in series, the charges have to push through both.

Right.

They do a certain amount of work to get through the first one, dropping some voltage, and then do the rest of the work getting through the second one, dropping the rest of the voltage.

The key is that the total voltage is shared between those two resistors in strict proportion to their resistance.

So if one resistor is much harder to get through, the charges have to expend more energy to cross it.

Exactly.

So a larger share of the total voltage drops across it.

Okay.

So we have a 10 volt supply and two identical resistors, say both 500 ohms.

The resistance is split 50 -50.

Which means the voltage is split 50 -50.

Five volts drop across the first.

Five volts drop across the second.

And if you attach your sensitive electronic component across just that second resistor,

you've successfully created a steady five volt supply from a 10 volt source.

The general equation for this is really straightforward.

The output voltage Vout equals the resistance of your chosen resistor, let's call it R2, divided by the total resistance of the circuit R1 plus R2.

Okay.

And then you multiply that fraction by the total input voltage V in.

Wait, so we are basically taking a fraction of the total resistance and multiplying it by the total voltage.

You've got it.

That proportional sharing is the heart of the potential divider.

Okay, but using fixed resistors means you are stuck with whatever voltage you initially designed for.

What if you need to adjust the voltage on the fly?

Then you use the practical variation from the textbook diagrams.

You replace the two fixed resistors with the single long variable resistor and you use a sliding contact.

Like a slider.

Right.

This is often called a potentiometer or a rheostat setup.

So picture a long coil of resistive wire.

The current enters at one end and exits at the other.

But instead of letting the current travel the whole way, you introduce a metal slider that touches the coil somewhere in the middle.

Exactly.

The current flows along the coil, hits the slider, and takes the shortcut out to your component.

By moving that slider up and down, you are physically changing the length of the wire the current has to travel through.

You are manually changing the ratio of the potential divider.

So when you turn the volume dial on a stereo or slide a dimmer switch on a dining room wall, you are physically dragging a metal contact across a resistor.

Yes.

You are smoothly adjusting V out from zero volts all the way up to the full V in.

You're changing the fraction of the total voltage that gets delivered to the speakers or the lights.

That is so intuitive.

And if we connect this to the bigger picture, a sliding contact is great for a human.

But what if we want the circuit to react to the environment automatically?

Oh, giving the circuit senses.

Exactly.

To do that, we replace one of the potential divider resistors with a sensor.

Like a light -dependent resistor or LDR.

The text goes into this in detail.

It does.

The LDR is a fantastic component.

Let's describe the graph from the text.

It's this exponential looking curve where the resistance drops dramatically as light intensity, which is measured in lux increases.

Right.

When an LDR is sitting in the pitch dark, its resistance is enormous.

It strongly opposes the flow of current.

But as photons of light hit the material, they transfer energy to the electrons inside, knocking them free and allowing them to participate in the current.

So as it gets brighter, the resistance just plummets.

Let's look at the worked example from the textbook to see how this behaves in a potential divider circuit.

Okay.

I'll walk us through the math.

So we have a circuit with a 10 volt supply.

10 volts.

Got it.

And it has a 3 .0 kilo ohm fixed resistor connected in series with an LDR.

Okay.

The text says that at a light level of 60 lux,

the graph shows the LDR has a 20 kilo ohm resistance.

So we're sharing the voltage in a three to 20 ratio.

Exactly.

So the total parts in our fraction is three plus 20, which gives us a total resistance of 23 kilo ohms.

And we want the voltage across the LDR.

So our V out equals 20 divided by 23.

Right.

And we multiply that fraction by our 10 volt supply.

Which gives you, well, the calculator will say 8 .69565 volts.

Right.

But the text gives a quick calculator tip here.

We need to round that to three significant figures.

So it becomes 8 .73 volts.

Perfect.

So during the day, if the light hits the LDR, its resistance is tiny compared to the fixed resistor.

So it claims a very small fraction of the total voltage.

But as the sun sets and it gets darker,

the LDR's resistance shoots back up to 20 kilo ohms or higher.

Suddenly it represents the vast majority of the total resistance.

So it claims a massive share of the available voltage.

And if you connect a street lamp relay across that LDR, the voltage will eventually rise high enough to trigger the relay and turn the lights on.

The circuit is responding dynamically to the environment.

It's brilliant.

And you can do the exact same thing with temperature using a component called a thermistor.

Right.

Specifically, the text

introduces negative temperature coefficient thermistors.

Yeah.

NTCs.

Yes.

So what does this all mean?

Why negative normal metal wires get more resistant when they get hot, right?

They do.

In a way.

And those vibrating atoms get in the way of the flowing electrons, causing more collisions, so the resistance increases.

Exactly.

But thermistors aren't made of standard metals like copper.

They are made from semiconductor materials.

Ah, semiconductors.

And semiconductors have a fundamentally different internal structure.

In a cold semiconductor, most of the electrons are tightly bound to their atoms.

There are very few free charge carriers.

So it acts almost like an insulator.

But when you apply heat, you are injecting thermal energy into the material.

That energy is enough to break those bonds, freeing massive numbers of electrons.

Wow.

Okay.

Yes.

The atoms are vibrating more, just like in copper.

But the sheer flood of newly freed electrons completely overwhelms the effect of the vibrating atoms.

It's like the traffic might be worse, but you've suddenly opened up 10 ,000 new lanes.

The electricity flows much easier and the resistance plummets.

Exactly.

So as temperature goes up, resistance goes down.

That's the negative coefficient.

But the textbook points out this relationship is not a straight line, right?

No.

The resistance temperature graph is an exponential decrease.

Which means if you're engineering a digital thermometer, you have to carefully calibrate it.

Right.

Because you can't just slap a linear scale on it like an old school class in mercury thermometer, where every degree is the exact same distance apart.

If you try to map a thermistor's voltage changes onto a straight, evenly spaced grid, your temperature readings will be wildly inaccurate.

The software has to be calibrated to the specific sloping curve of that semiconductor.

Which brings us to a profound underlying paradox that haunts every single measurement we've discussed so far today.

It's a massive roadblock.

We've been talking about using voltmeters to measure these potential dividers or measuring the terminal voltage of a battery defined as internal resistance.

But we established earlier that drawing any current from a battery forces the charges to do work against the internal resistance.

Right.

Which creates lost volts, which lowers the voltage you're trying to measure in the first place.

Exactly.

It's the electronics version of the observer effect.

The very act of looking at the voltage changes the voltage.

Even a high end digital voltmeter draws a tiny bit of current to function.

So how could we ever measure the true uncorrupted electromotive force of a cell?

To measure the perfect EMF, you must draw absolutely zero current.

Not a small current.

Zero current.

And doing that requires moving away from digital multimeters entirely and returning to a brilliantly simple analog circuit.

The potentiometer.

Yes.

It is a device for comparing potential differences without drawing any current from the test cell.

Okay.

Paint a visual picture of this circuit set up for us because it's so clever.

All right.

You start with a reliable power supply.

We call this the driver cell.

Let's say it's EMF is E naught.

Okay.

E naught.

You connect this driver cell across a long bare stretch of highly uniform resistance wire.

Usually this wire is exactly one meter long, stretched tight across a wooden board.

So from point A to point B.

Right.

Let's say the driver cell provides 2 .0 volts.

The current flows out of the cell, enters the bare wire at point A, travels the full meter and exits at point B.

And because the wire has uniform resistance, the voltage drops perfectly steadily along that entire meter.

Exactly.

Point A is at 2 .0 volts.

At the 50 centimeter mark, the voltage has dropped exactly in half to 1 .0 volt.

At the end, point B, it has dropped to zero volts.

The wire acts as a physical spatial ruler for voltage.

Now we bring in the mysterious cell we want to measure.

The unknown cell, which we'll call EX.

Right.

We connect its positive terminal to the exact same point A on the wire.

Okay.

So both positive terminals are at point A.

Then we take a wire from its negative terminal, run it through a highly sensitive galvanometer.

Which detects even microscopic currents.

And end that wire with a metal tool called a jockey.

The jockey.

It has a sharp metal edge, allowing you to make precise electrical contact at any point along that one meter resistance wire.

So when you touch the jockey to the wire, you are essentially setting up a tug of war.

The driver cell is trying to push current down the wire, and the unknown cell is trying to push its own current through the galvanometer.

If you touch the jockey near point A, where the driver cell's voltage is very high, the driver cell wins the tug of war.

Right.

It forces current backward through the unknown cell and the galvanometer needle swings to the left.

But if you touch the jockey near point B, where the driver's voltage is mostly dissipated, the unknown cell wins.

It pushes current out and the needle swings to the right.

But if you carefully test different spots along the wire, you will eventually find one specific magical location.

That's balance points.

Let's call it point Y.

Yes, point Y.

At this exact physical location, the voltage dropping across the wire perfectly matches the electromotive force of the unknown cell.

The tug of war is perfectly balanced.

Neither side can push a single electron.

The galvanometer needle sits perfectly still at dead zero.

This is the null method, and it is the key to defeating internal resistance.

Exactly.

If the needle reads zero, it means zero current is flowing out of the unknown cell.

And if zero current is flowing, the I in our equation V equals E minus IR is zero.

The lost volts drop to zero.

The terminal potential difference at that exact moment is identical to the true perfect EMF.

You have successfully measured the voltage without drawing any current.

It's genius.

It is.

But there is a massive practical trap here that ruins this experiment for people all the time.

Oh, the textbook has a crucial warning about this.

When you are moving the metal jockey along the bare wire to find that balance point, you must tap it gently.

You never ever slide it.

Sliding is catastrophic.

If you drag a sharp metal edge along a bare wire, you act like a cheese grater.

You scrape off microscopic layers of metal.

And if the wire becomes physically thinner in some places than others?

Its cross -sectional area changes, which means its resistance changes.

It is no longer uniform.

The voltage will no longer drop smoothly and evenly across that meter.

Your physical ruler becomes warped, and the proportionality the entire experiment relies on is completely destroyed.

So tap, don't slide.

Exactly.

Now, assuming your wire is pristine and found your balance point at y, calculating the unknown EMF is just a matter of ratios.

Let's hear the key equation.

The unknown EMF, ex, equals the length of the wire from point a to point y divided by the total length of the wire from a to b.

So length ay over length ab.

That gives you a fraction.

Right.

And you multiply that fraction by the driver cell's voltage, e -naught.

It's so clean.

Ex equals parenthesis ay over ab, close parenthesis, times e -naught.

Though, to be perfectly precise, the driver cell itself has internal resistance, so we don't always know its exact voltage.

Oh, right.

So how do we bypass that?

Engineers use the potentiometer to compare two different cells.

You find the physical balance point on the wire for a cell with a perfectly known EMF.

Let's call its balance length AC.

Okay.

Then you swap in your unknown cell and its new balance length AD.

The ratio of their physical lengths on the wire is exactly equal to the ratio of their voltages.

So ex over ey equals length AC over length AD.

You don't even need to know what the driver's cell is doing.

It is an incredibly elegant workaround to a deeply stubborn physical problem.

It really is.

And it perfectly encapsulates the journey we've taken today.

Yeah.

We started by acknowledging the messy reality that physical materials inherently resist the flow of electricity, forcing us to calculate lost energy.

Right.

The internal resistance.

And then we explored how to harness that resistance to our advantage using potential dividers to slice and share voltage proportionally.

Giving components exactly what they need.

And we brought those circuits to life by integrating semiconductor sensors like LDRs and NTC thermistors.

Allowing resistance to fluctuate dynamically with changes in light and heat.

And finally, we engineered a way to measure these systems flawlessly using the geometry of a wire and the null method to outsmart the physics of internal resistance.

It is a profound sequence of problem solving.

It makes you realize that mastering electricity isn't just about memorizing equations.

It's about deeply understanding how the physical world pushes back and how to engineer around it, which brings us right back to that Baghdad artifact.

This raises an important question.

If ancient people thousands of years ago were truly using a chemical reaction to drive an electrical current,

they were dealing with these exact same physical realities.

Yeah.

The vinegar, the copper, the iron, it all had internal resistance.

They had lost volts.

They had to design their system to account for it, even without the modern mathematical language we use today.

It makes you wonder if they were intuitively engineering around electromotive force to plate gold onto jewelry in the ancient world, what other profound physical principles might be hiding out there in the archeological record?

Just waiting for us to figure out the circuit they were trying to build.

That is a truly fascinating thought to leave on.

It forces us to reconsider the timeline of human ingenuity, doesn't it?

Physics is the fundamental operating system of the universe, and we are just the latest generation trying to write code for it.

Beautifully said.

Well, thank you to everyone listening for diving into the mechanics of practical circuits with us.

The messy physical realities of electricity make it challenging, but they are also what make it possible to build the modern world.

Absolutely.

From all of us here at the Last Minute Lecture Team, thank you so much for joining us on this deep drive.

Keep questioning, keep learning, and we will catch you next time.