# Ampère’s Law: Crash Course Physics #33

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.

## 100 thoughts on “Ampère’s Law: Crash Course Physics #33”

• ### scienceandmind Post author

Thanks Ampere!

• ### Ezz Eldin Post author

Arabic translation bls

• ### Regina Falangee Post author

Can 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 Morrison Post author

Isn'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 Allan Post author

It's distractingly obvious that she's reading a teleprompter.

• ### Pilot Owl Post author

my 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?

• ### NightLurk Post author

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

• ### Chanandler Bong Post author

For 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.

• ### Jack Burton R-1 Post author

okay…got it thanks.

• ### Joshua Weickum Post author

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

• ### Mark Holm Post author

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

• ### Gernuts Post author

Joke's on you. My laptop has passive cooling 😛

Great video!

• ### Hemanth V. Alluri Post author

Just 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 Diller Post author

she is so damn good looking it's not even funny 🙁

• ### VoltZ Post author

quantum physics, general relativity, or gravity for einstein

• ### Marek Post author

Great 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 Tyagi Post author

I did not get why they attracted each other.

• ### Rocketeerian Post author

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

• ### Abhishek Singh Post author

Great animation..and explanation. Well done.

she 's hot

• ### Shivam Chaudhary Post author

I love ur t-shirt. by the way ur way to express is quite awesome

• ### Mr Yohan Post author

Hey 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 Jangid Post author

u luks hoT..😍😍…and thnks..for this video..

• ### Jeb Bush Post author

Damn I'm starting to like physics with this person

• ### BK udonnoodles Post author

Gosh 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 Hamas Post author

Just brilliant, totally brilliant loved the teaching method and the graphics.

• ### Mr. Whiskers Post author

lets AMP it up

• ### Doga Tahan Post author

I really did not understand the attraction of the wires and the axis rotation of the wire

• ### Ross Milligan Post author

They 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 dau Post author

instead of the third right hand rule use a left hand rule instead

• ### Suvraneel Bhuin Post author

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

• ### Edwin Ramirez Post author

excellent. thank you

• ### Abjo Das Post author

Awesome animations! Really helps to build the concept. Thanks a ton

very nice

• ### Rafia Shahzad Post author

should be in american accent..I feel so difficulty in understanding british accent

• ### BACHAN SINGH Post author

why 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 Rahman Post author

HANK GREEN SHOULD TEACH ALL COURSES

• ### Dave Jones Post author

You talk way to fast so a thumbs DOWN

Prove ΔxH=J

• ### amr elkot Post author

please i need all the videos translated into Arabic

• ### Ron Bessler Post author

The 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 CF Post author

So ampere's law is just calculus

• ### Hady Humanism Post author

Wonderful but speaking speed is very high

• ### Cole Hatmaker Post author

Maybe 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 Raju Post author

she 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 Raju Post author

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

• ### Abishek Raju Post author

Could guys do some videos on Music Theory??

You 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 Sumi Post author

I 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 Karunarathne Post author

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

• ### sunil negi Post author

O my God.!!! she is so beautiful..!

• ### Nadeesha Chathumal Post author

Nice explanation with animations..

• ### Jason Wills Post author

Thisisbrilliant

• ### fahim ahmed Post author

you talk way too fast

• ### Hallow! Post author

The parallel wire animation has a mistake!

• ### Julesdoesstuff Post author

Who else is screwed on physics c tomorrow

• ### Mina Post author

So useful!!!❤

• ### bhanupratap yadav Post author

You explain so well. Lots of love

• ### Phasor Systems Post author

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

Woah!!

• ### Obaid Rahman Sabawoon Post author

why is she in a rush ?? too fast.

• ### Tomas Molina Post author

Even a dog could have understanded that integration explanation

• ### Fernando Cadena Post author

Amazing lesson! Congratulations from Brazil!

• ### AMEER ALAM Post author

which software used by u for edition video

• ### Navtej Singh Post author

No words for you mam u r best in physics

• ### Hong Vincent Post author

I 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 Ranjan Post author

Behen thoda there bol yrr humari habbit nhi itti fast english ki

• ### Ryan Post author

unless 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.

Good

• ### Armando Bustamante Post author

Does this girl have an instagram?

• ### Brendan Greenberg Post author

CrashCourse, 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 Anzoategui Post author

Honestly 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

• ### Camille COLLET Post author

The force called Laplace force

• ### Mehul Arora Post author

how 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 ?

• ### vatt Post author

I wanna put my magnetic field into your coil baby

bullet speed

• ### Anurag Hooda Post author

At 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 Ahmed Post author

why 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.

TOOO FAST??

• ### Amrita S Post author

Her accent is more important than the topic. shrugs

• ### Tayyab Zaman Post author

Miss home tuition doo gi😂😂😍

• ### Daniel Machuca Post author

Crash 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 OConnor Post author

Easy to listen to and easy on the eyes.

• ### Soul of Zion Post author

Full of animation errors!! thanks though!

• ### sada que Post author

Thanks from Perú

• ### Super Nova Post author

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

• ### Amaan asad Khan Post author

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

she' pretty

thanks sis

Ídolo!

• ### Denver Maburutse Post author

the 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 Prada Post author

explination is so fast

• ### JAGUAR 007 Post author

Love form india