# Ampère’s Law: Crash Course Physics #33

Articles, Blog 100 Comments

It was the autumn of 1820. Hans Christian Oersted had just discovered

the connection between electricity and magnetism. Meanwhile, a French physicist named André-Marie

Ampère was experimenting with some wires, trying to learn more about the connection between currents and the magnetic fields they create. He took two parallel wires, ran a current through both of them in the same direction, and the wires attracted each other! And when he ran a current through both wires in the opposite directions, the wires repelled each other! Studying this strange force between the wires led Ampère to discover one of the most fundamental laws of electromagnetism: what we now call Ampère’s Law. And that’s not the only weird thing that current-carrying

wires do. If you wrap a current-carrying wire into a

coil, the inside of the coil acts like a magnet. There’s a north pole at one end of the coil

and a south pole at the other. And if you put a loop of current-carrying

wire in a magnetic field, it’ll turn! Ampère’s law lets you calculate the strength of the currents and magnetic fields in all these situations. This also helps explain how motors work. [Theme Music] So first, those two parallel wires: why do

they attract and repel each other? It’s easier to see why the two wires act the

way they do if you look at one wire first. Like we talked about in our last episode, the current running through a wire generates a magnetic field. So, let’s say you have a long, straight wire

with a current running through it. The current will create a magnetic field circling

the wire. That magnetic field decreases the further

you are from the wire. If you draw a circle that’s, say, a centimeter from the wire, the magnetic field along the circle will have a set strength. Ampère realized that the stronger the current is that’s running through the wire, the stronger the magnetic field would be along that circle. That’s the basic logic behind Ampère’s Law. But this is physics, and in physics, we tend

to express relationships in terms of equations. The equation for Ampère’s law applies to any kind of loop – not just a circle – surrounding a current, no matter how many wires there are or how they’re arranged or shaped. The law is valid as long as the current is

constant. The equation itself says that the integral of the magnetic field, B, along the loop, times the cosine of theta, with respect to distance, is equal to a constant – called mu_0 – multiplied by the current running through the loop. This equation just means that the total magnetic field along the loop is equal to the current running through the loop, times a constant number. The constant mu_0 is sometimes called the magnetic constant, and it’s equal to 4 times pi times 10to the -7th Newtons per Amperes squared. Now, you may have noticed that there’s an integral on the left-hand side of the equation for Ampere’s law. And you might remember that we use integrals when we need to add up lots of infinitely tiny values. Well, in Ampère’s Law, we’re adding up all the

little bits of magnetic field along the loop. We’re saying that all those bits of magnetic field added together are equal to the enclosed current, times the magnetic constant. B is the strength of the magnetic field at

each point along the loop. Theta is the angle between the magnetic field

and each point on the loop. And ds is referring to each infinitely tiny

section of the loop. The mathematics of Ampère’s law can get very

complicated very quickly. But to get a basic sense of how it works, let’s return to our scenario: a circle around one long straight wire. We’re trying to find the magnetic field at each point on the circle – that’s B – in terms of the enclosed current and the radius of the circle. So first, let’s solve the integral in Ampère’s law,

to get the total magnetic field along the circle. According to the law, we’re solving the integral of the magnetic field, times the cosine of theta, with respect to the points along the circle. But we can simplify this integral pretty easily. First, you’ll notice that the magnetic field coming from our wire is parallel to the circle at every point. So the angle, theta, is 0, and the cosine

of 0 is 1. Anything times 1 is equal to itself, so we can just knock the cosine of theta term out of the integral. Now we’re left with the integral of the magnetic field, B, with respect to the points along the circle. But every part of the circle is the exact

same distance from the wire! So the magnetic field will be the same at

every point. In other words: B is constant, so we can move

it in front of the integral sign. Now all we need to do is figure out the integral of all the points along the circle, which is equal to the circumference of the circle. So, 2 times pi times the radius. Putting that all together, we find that when we apply Ampère’s law to a long straight wire, the total magnetic field along a circle surrounding a wire is equal to B times 2 times pi times the radius. And that total magnetic field is equal to

the magnetic constant times the enclosed current. So! For a long straight wire, B is equal to the magnetic constant times the enclosed current, divided by 2 pi r. The equation for the magnetic field along a circle surrounding one wire turned out to be really important for Ampère when he was trying to figure out what was going on with two wires. When both wires had current running through them in the same direction, they attracted each other. And when the current was going in opposite

directions, they repelled each other. It’s easy to see why, if you apply the first

right-hand rule. That’s the one that says if you point your right thumb in the direction of a current and curl your fingers, the magnetic field points in the same direction as your fingers. So first, let’s look at the wires with currents

running in the same direction. For this example, we’ll say that they’re vertical

wires, with the current flowing upward. If you point your right thumb in the direction of the current in each wire, your fingers will curl in the direction of the magnetic field. The magnetic field from the wire on the left

will be pointing to the right. And the magnetic field from the wire on the right will be pointing to the left, so the wires will attract each other. For the case where the current is flowing in opposite directions, the reverse is true, so they’ll repel each other. Now, Ampere also wanted to find the force

from the magnetic field on the wires. Like we talked about last time, the force depends on the angle between the current and the magnetic field, the strength of the current, the length of the wire, and the strength of the magnetic field. Calculating that magnetic field, B, was the

tricky part. But the equation he came up with, the one that we now call Ampere’s Law, allowed him – and future physicists! – to figure out what B was in a lot of situations, including the case of the two parallel wires. So the two parallel wires attracted and repelled each other because of the magnetic field created by the current. What about the coil of wire that turned into

a magnet? Well, you can probably guess that its behavior also has to do with the magnetic field produced by a current. See, that coil of wire is a special shape

called a solenoid. And when a solenoid has a current running through it, it produces a magnetic field, basically all of which goes through the inside of the coils. If you curl your right hand around the solenoid so that your fingers point in the direction of the current running through the loops, your thumb will point in the direction of the magnetic field. Ampère’s law is useful for solenoids, too: it says that the magnetic field inside the coils, B, is equal to the magnetic constant, times the current running through the coils, times the number of coils. So that’s what happens when loops of wire

create a magnetic field. When you stick a loop of wire in a magnetic

field, something a little stranger happens:

the loop of wire turns. That’s because the magnetic field creates

a torque on the wire. Take a look at this loop of wire. The horizontal parts of the loop are parallel to the magnetic field, so it won’t exert a force on them. But the vertical parts of the loop are perpendicular to the magnetic field, so it will exert a force on them – a force that turns the loop. From the last episode, we know that the force from the magnetic field on the wire will be equal to the current, times the length of that part of the coil, times the magnetic field. And we can use the second right-hand rule

to figure out the direction of that force. If you point your hand in the direction of the current, then bend your fingers in the direction of the magnetic field, your thumb will point in the direction of the force. Which turns out to be away from you for the left-hand side of the coil, and toward you for the right-hand side. So the coil turns clockwise. This is how electric motors work: they have an electric current that continuously flips directions, making loops of wire spin. Those moving loops of wire can be used to do mechanical work, like turning the drum in your washing machine, or your power drill, or the fan that probably cools your computer. There are electric motors all over the place. So the next time you wash your clothes, or put together some furniture, or use your computer without it overheating, or do anything else that involves an electric motor, you have Ampère to thank. Today, you learned about Ampère’s law, and

how it applies to a long straight wire. We also talked about the forces between two parallel wires, and the magnetic field created by a solenoid. Finally, we described the torque on a current

loop. Crash Course Physics is produced in association

with PBS Digital Studios. You can head over to their channel and check out a playlist of the latest episodes from shows like: Gross Science, PBS Idea Channel, and It’s

Okay to be Smart. This episode of Crash Course was filmed in

the Doctor Cheryl C. Kinney Crash Course Studio with the help of these amazing people and our

equally amazing graphics team, is Thought Cafe.

scienceandmindPost authorThanks Ampere!

Ezz EldinPost authorArabic translation bls

bubblesPost authorHey could y'all please make a youtube video about greek mythology?

Regina FalangeePost authorCan I just say after how long I've been waiting for crash course physics vids and then to discover this amazingness! finally!!! Also delighted I like the host! I'm thouroughly pleased… ¦-)

Colby MorrisonPost authorIsn't it better to describe it as a line integral instead of some simplification of the dot product? You can then see what you're really doing, "adding up" all the parts of the magnetic field through the surface. It makes much more intuitive sense that way, if we "add up" all the magnetic field around the curve, we get the enclosed current times mu-naught.

Christopher AllanPost authorIt's distractingly obvious that she's reading a teleprompter.

Pilot OwlPost authormy laptop doesn't have a fan 🙂

also the original experiment is using DC current am I correct?

The electric motors on the other hand use AC motors?

NightLurkPost authorMy little head enjoyed watching this despite my big head constantly telling it to quit it out…

Chanandler BongPost authorFor everyone that watches Crash Course and finds it too complicated, these videos just give a very brief summary of physics topics. If you actually want to understand in depth, buy a textbook or also visit places like Khan Academy or Isaac Physics.

angel 11Post authorCould you add Arabic subtitle?

Jack Burton R-1Post authorokay…got it thanks.

Joshua WeickumPost authorYou should do some books on tape, your voice is easy on the ears. Youre not bad to look at either.

Mark HolmPost authorThe animation at 5:24 also gets the right hand rule backwards, as well as the one just previous.

GernutsPost authorJoke's on you. My laptop has passive cooling 😛

Fu TalksPost authorGreat video!

Hemanth V. AlluriPost authorJust a heads up, but you really should have used dl in the equation as opposed to ds because otherwise what you could confuse viewers with the Gauss law for magnetism in which case the surface integral of B.dS (sometimes also shown as B.dA) should give 0. dl because we are considering a differential element of length of the ampere loop. Correct me if you disagree though.

Murrant DillerPost authorshe is so damn good looking it's not even funny 🙁

VoltZPost authorquantum physics, general relativity, or gravity for einstein

MarekPost authorGreat video! But you got the second hand rule wrong. 7:10 You should point your hand (I prefer index finger) to the direction of CHARGE VELOCITY and not to the current flow direction. Since inside the wire electrons are moving in opposite direction to the current flow, you should point your index finger in the opposite direction than you showed in the video :). This way your example with two wires will make sense.

Apoorv TyagiPost authorI did not get why they attracted each other.

RocketeerianPost authorWhen you add vector calculus to this… The difficulty to understand this just blows up! lol

Abhishek SinghPost authorGreat animation..and explanation. Well done.

Lila StevensonPost authorshe 's hot

Shivam ChaudharyPost authorI love ur t-shirt. by the way ur way to express is quite awesome

Mr YohanPost authorHey Crash Course Physics!

The direction of the magnetic field at 5:15 is wrong.

May you please make sure to correct it with a notice on the video?

Thanks.

Krishna JangidPost authoru luks hoT..😍😍…and thnks..for this video..

Jeb BushPost authorDamn I'm starting to like physics with this person

BK udonnoodlesPost authorGosh having a visual representation is soo much easier than figuring out from equations. Also make me realized why B is assumed constant for all ds and thus can be taken out of integral.

Muzammil HamasPost authorJust brilliant, totally brilliant loved the teaching method and the graphics.

Mr. WhiskersPost authorlets AMP it up

Doga TahanPost authorI really did not understand the attraction of the wires and the axis rotation of the wire

Ross MilliganPost authorThey really need to add an annotation at 5:14 explaining that the arrows of field direction are wrong according to the right-hand rule, really messed me up for a second there.

atte dauPost authorinstead of the third right hand rule use a left hand rule instead

Suvraneel BhuinPost authorHey! that torque on current loop part is completely wrong…. That loop must turn in backwards direction… Isn't it?¿?

Edwin RamirezPost authorexcellent. thank you

Abjo DasPost authorAwesome animations! Really helps to build the concept. Thanks a ton

Muhammad AbubakarPost authorYour speed is too high

Dinesh RaiPost authorvery nice

Rafia ShahzadPost authorshould be in american accent..I feel so difficulty in understanding british accent

BACHAN SINGHPost authorwhy are you speaking so fast .. no one unknown to thin topic can't learn through you because time is required to think and understand. .. you should speak a bit slower 😐😐😐

Shynan RahmanPost authorHANK GREEN SHOULD TEACH ALL COURSES

Dave JonesPost authorYou talk way to fast so a thumbs DOWN

Ritesh KalpandePost authorProve ΔxH=J

amr elkotPost authorplease i need all the videos translated into Arabic

Ron BesslerPost authorThe videos explained and animated wonderfully! But it is to fast to actually understand. I personally see these videos in a slow motion mode and still find it not easy to understand. I know you wanted as to see it in 10 min but maybe it would be more useful to have longer videos with a deeper understanding. Honestly it would be a dream come true to have 40min video on a subject with examples nice pace of talking and your wonderful explanations and animations.

FRTZ CFPost authorSo ampere's law is just calculus

Hady HumanismPost authorWonderful but speaking speed is very high

Cole HatmakerPost authorMaybe I’m missing something, but the description of theta, in the integral side of the equation in Ampere’s law, as the angle between the magnetic field and the current doesn’t seem to make sense. Isn’t it the angle between the magnetic field and any given point on the loop surrounding the wire?

Abishek RajuPost authorshe kinda turns me on when i play it at x0.75 playback speed. I'd rather watch it at the normal speed and not get distracted.

Abishek RajuPost authortry pausing an taking notes for those who find the video to be kinda racing. It really helps.

Abishek RajuPost authorCould guys do some videos on Music Theory??

Raiyad RaadPost authorYou speak too fast and the animations making it difficult to understand. The colors hurt my eyes and the change of animations were too fast I think.

Vikibe K SumiPost authorI just can't say how thankful i am to make me understand. All other videos I saw was a thrash untill I saw this

Senawirathne KarunarathnePost authorthe motor effect animation isn't spinning in the correct way. thankz for the hard work though

sunil negiPost authorO my God.!!! she is so beautiful..!

Nadeesha ChathumalPost authorNice explanation with animations..

Jason WillsPost authorThisisbrilliant

fahim ahmedPost authoryou talk way too fast

Hallow!Post authorThe parallel wire animation has a mistake!

JulesdoesstuffPost authorWho else is screwed on physics c tomorrow

MinaPost authorSo useful!!!❤

bhanupratap yadavPost authorYou explain so well. Lots of love

Phasor SystemsPost authorHaving problems with circuits? Try circuit solver Bump into: 'Circuit Solver' by Phasor Systems on Google Play.

Richer XDPost authorWoah!!

Obaid Rahman SabawoonPost authorwhy is she in a rush ?? too fast.

Tomas MolinaPost authorEven a dog could have understanded that integration explanation

Fernando CadenaPost authorAmazing lesson! Congratulations from Brazil!

AMEER ALAMPost authorwhich software used by u for edition video

Navtej SinghPost authorNo words for you mam u r best in physics

Hong VincentPost authorI am really glad you did all these videos. I really wished i had access to all of these when i was a student. Kudos to you, and subscribed!

Hare Ram RanjanPost authorBehen thoda there bol yrr humari habbit nhi itti fast english ki

RyanPost authorunless you mean electron flow direction (which would only complicate stuff) the field should be turning in the opposite directions. It's called right hand rule because you use your right hand, Seems like you used your left.

AANAND KUMARPost authorGood

Armando BustamantePost authorDoes this girl have an instagram?

Brendan GreenbergPost authorCrashCourse, it has been almost 2 years since this video was posted and you still don't have an annotation explaining that the magnetic fields are in the wrong direction at 5:17 and 5:25. I spent a lot of time trying to figure out how you were getting those directions before I realized that it was a mess up in the video. Please fix this.

Jesus AnzoateguiPost authorHonestly everything that came out of her mouth after the mistake at 5:14 I couldn’t even believe her , correct the mistake you’re just losing credibility

Ahsan TanveerPost authorPlease speak slowly

Camille COLLETPost authorThe force called Laplace force

Mehul AroraPost authorhow am i supposed to learn amperes law from this series of crash course .I m not able to concentrate on the topic .Why she is so beautiful ?

vattPost authorI wanna put my magnetic field into your coil baby

Naveen KumarPost authorbullet speed

Anurag HoodaPost authorAt 2:08 it's written that theta is the angle between current and magnetic field, shouldn't it be the angle between the element ds and the magnetic field?

Jamil AhmedPost authorwhy you make things easy. taught us as our tutors did.(4 Year)but i respect them still. I understand now the Ampere's law in a good and effective way.

Johan Manoj MathewPost authorTOOO FAST??

Amrita SPost authorHer accent is more important than the topic.

shrugsTayyab ZamanPost authorMiss home tuition doo gi😂😂😍

Daniel MachucaPost authorCrash Course, I believe there is a problem with the direction of which the magnetic field is going at 5:16. If the current of the wire is pointing up, shouldn't the magnetic direction be pointing counterclockwise? I hope you can fix this. Thanks

Peter OConnorPost authorEasy to listen to and easy on the eyes.

Soul of ZionPost authorFull of animation errors!! thanks though!

sada quePost authorThanks from Perú

Super NovaPost authorU r talking so fast for me i am not a native English speaker! Moreover the Arabic translation is bad

Amaan asad KhanPost authorShe's so distracting to look at she's so beautiful, why can't she just do a voice over the animation

MrNegrete7Post authorBad teacher.

Mohit malaniPost authorstarted following your channel and your dressing style.

SHIVAM THAPAPost authorshe' pretty

shawn shawnPost authorthanks sis

Mario tPost authorÍdolo!

Denver MaburutsePost authorthe theta is first mentioned as the angle between the B-field and current (2:09) and then later on mentioned as the angle between B-fields at different points of the loop. I don't know which one is correct

Shubha PradaPost authorexplination is so fast

JAGUAR 007Post authorLove form india