Ampere's circuital law (with examples) | Moving charges & magnetism | Physics | Khan Academy

Summary

In this educational video, the concept of Ampere's Circuital Law is introduced as a counterpart to Gauss's Law in magnetism. The law is explained through a practical example involving three wires carrying currents. The video demonstrates how to calculate the magnetic field by taking the dot product of the magnetic field and a tiny length vector 'dl' around a closed loop. It emphasizes that the law applies only to closed loops and the result of the integral is the product of the permeability constant and the total current enclosed by the loop. The video also explains how to determine the direction of positive current using the right-hand rule and highlights the importance of attaching a surface to the loop to identify the enclosed current, a concept that becomes crucial in more complex scenarios.

Takeaways

• 🧲 Ampere's Circuital Law is the magnetic equivalent to Gauss's Law for electric fields, used to calculate magnetic fields in symmetric situations.
• 🔁 The law is applied by drawing a closed loop in space and calculating the line integral of the magnetic field (B) dot product with an infinitesimal length vector (dl) around the loop.
• ⚖️ The integral of B dot dl around a closed loop equals the product of the permeability of free space (μ₀) and the total current (I) enclosed by the loop.
• 🤔 The direction of the loop and the enclosed current can be determined using the right-hand rule, which aligns the thumb with the direction of positive current.
• 🔄 The enclosed current (I_enclosed) is defined as the total current passing through a surface attached to the loop, which can be visualized as a soap film.
• 📐 The shape of the loop and the attached surface can be any shape, including flat or blown into an open shape, but the enclosed current remains the same.
• ⚠️ Ampere's Law is only applicable for closed loops; it does not work for open paths.
• 🔗 The magnetic field considered in Ampere's Law is the total magnetic field due to all currents, both enclosed and non-enclosed by the loop.
• 🔄 The contributions of B dot dl by non-enclosed currents often cancel out, similar to how electric flux depends only on enclosed charges in Gauss's Law.
• 🔍 In complex scenarios, attaching a surface to the loop helps in determining the enclosed current, which might not be immediately obvious.

Q & A

What is Ampere's Circuital Law?

Ampere's Circuital Law, also known as Ampere's Law, is a fundamental law of electromagnetism that relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is used to calculate the magnetic field created by electric currents.

How does Ampere's Law relate to the magnetic field and a closed loop?

According to Ampere's Law, the closed line integral of the magnetic field (B) around a closed loop is equal to the product of the permeability of free space (μ₀) and the total current (I) enclosed by the loop.

What is the significance of the dot product in Ampere's Law?

The dot product in Ampere's Law signifies the component of the magnetic field that is tangential to the path of integration along the closed loop. It ensures that only the magnetic field components along the direction of the loop contribute to the integral.

Why is it necessary to use a closed loop in Ampere's Law?

A closed loop is necessary in Ampere's Law because the law is based on the concept of a circulation of the magnetic field around a current. An open path would not allow for a complete circulation, and thus, the law would not hold.

What does the enclosed current in Ampere's Law represent?

The enclosed current in Ampere's Law represents the total current that passes through any surface bounded by the closed loop. This is determined by using the right-hand rule to determine the direction of positive current with respect to the direction of traversal around the loop.

How is the direction of the magnetic field related to the direction of the current in Ampere's Law?

The direction of the magnetic field is related to the direction of the current through the right-hand rule. If you curl the fingers of your right hand in the direction of the current, your thumb points in the direction of the magnetic field.

What is the role of the surface attached to the loop in Ampere's Law?

The surface attached to the loop, often imagined as a soap film, helps to visualize and calculate the enclosed current. The current that penetrates this surface is considered the enclosed current, which is used in the calculation of the magnetic field using Ampere's Law.

Can Ampere's Law be applied to non-symmetric situations?

While Ampere's Law is particularly useful in symmetric situations due to its simplicity, it can be applied to non-symmetric situations as well. However, the calculations may be more complex and require a deeper understanding of the integration of the magnetic field around the loop.

How is Ampere's Law different from Biot-Savart Law?

Ampere's Law is an integral law that calculates the total magnetic field around a closed loop due to all currents enclosed by the loop, while Biot-Savart Law is a differential law that calculates the magnetic field due to a small segment of current. Ampere's Law is often more convenient for symmetric situations.

Why does the magnetic field from non-enclosed currents not contribute to the integral in Ampere's Law?

In Ampere's Law, the magnetic field contributions from non-enclosed currents tend to cancel out due to their opposite directions as you go around the loop, resulting in a net contribution of zero. This is similar to how electric flux through a Gaussian surface is contributed only by the enclosed charge.

Outlines

00:00

🧲 Introduction to Ampere's Circuital Law

The paragraph introduces Ampere's Circuital Law, which is analogous to Gauss's Law in electricity but for magnetism. It explains that Ampere's Law helps in calculating magnetic fields in symmetrical situations. The law is demonstrated through an example involving three wires carrying currents i1, i2, and i3. A closed loop is drawn in space, and the law instructs to find the dot product of the magnetic field and a tiny length vector (dl) at every point along the loop. The summation of these dot products, represented by an integral, is then calculated over the entire loop. The integral of B dot dl around a closed loop equals μ₀ times the total current enclosed by the loop, a concept that will be explored further in the video series.

05:02

🔗 Calculating Enclosed Current with Ampere's Law

This section delves into the concept of 'enclosed current' in Ampere's Law. Ampere suggests visualizing the loop with an attached surface, like a soap film, to determine the enclosed current, which is the total current passing through the surface. The example given involves calculating the magnetic field for a rectangular loop, where only certain currents (i2 and i3) pass through the surface. The right-hand rule is used to determine the direction of positive current. The paragraph also touches on the equivalence of Ampere's and Biot-Savart's laws and the importance of considering only enclosed currents on the right side of Ampere's Law, while the magnetic field on the left side is due to all currents, both enclosed and non-enclosed.

10:03

🌀 Surface Attachment and Complex Situations in Ampere's Law

The final paragraph discusses the practical application of attaching a surface to the loop in Ampere's Law, which aids in determining the enclosed current, especially in complex scenarios. It mentions that the surface can be any shape, not just flat, and that the enclosed current remains the same regardless of the surface's form. The paragraph concludes by emphasizing the importance of this surface attachment technique, which will become more evident in future examples involving more complex magnetic field calculations, such as those involving capacitors.

Keywords

💡Coulomb's Law

Coulomb's Law is a fundamental principle in physics that describes the electrostatic interaction between electrically charged particles. It states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. In the video, Coulomb's Law is mentioned as a comparison to Ampere's Law, highlighting how both laws help calculate fields due to point charges or currents, but in different contexts.

💡Magnetic Field

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. The video script discusses how magnetic fields are calculated due to point current elements using Ampere's Law, which is analogous to how electric fields are calculated using Coulomb's Law for point charges.

💡Gauss's Law

Gauss's Law is one of Maxwell's equations in electromagnetism, which relates the electric flux through a closed surface to the enclosed electric charge, expressed as the product of the electric constant and the charge enclosed. The video mentions Gauss's Law in the context of calculating electric fields in symmetric situations and draws a parallel with Ampere's Law for magnetic fields.

💡Ampere's Circuital Law

Ampere's Circuital Law, also known as Ampere's Law, is a fundamental law of electromagnetism that relates the integrated magnetic field around a closed loop to the electric current passing through the loop. The video script uses this law to explain how magnetic fields can be calculated for symmetric situations, emphasizing its utility in simplifying calculations.

💡Closed Loop

In the context of Ampere's Law, a closed loop refers to a path in space that does not intersect itself and has no endpoints, allowing for the calculation of the magnetic field around it. The video script instructs to draw a closed loop anywhere in space to apply Ampere's Law, highlighting the flexibility in choosing the path for the calculation.

💡Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In the video, the dot product is used to calculate the component of the magnetic field that is tangent to the closed loop, which is essential for applying Ampere's Law.

💡Integral

In calculus, an integral is a mathematical concept, the result of an integration, which is the inverse of the derivative. The video script mentions the integral as the method to sum up the dot product of the magnetic field and the tiny length vector (dl) over the entire closed loop, which is a key step in applying Ampere's Law.

💡Enclosed Current

The enclosed current in Ampere's Law refers to the total current that passes through a surface bounded by the closed loop. The video explains how to determine the enclosed current by imagining a surface attached to the loop, such as a soap film, and calculating the current that 'punches through' this surface.

💡Right Hand Rule

The right-hand rule is a common mnemonic for understanding the cross product and the direction of the magnetic field around a current-carrying conductor. In the video, the right-hand rule is used to determine the positive direction of the enclosed current relative to the direction of the loop, which is crucial for correctly applying Ampere's Law.

💡Magnetic Field Due to Enclosed Currents

The magnetic field due to enclosed currents is the total magnetic field calculated using Ampere's Law, which only considers the currents that pass through a surface bounded by the closed loop. The video script clarifies that while the enclosed current is considered on one side of the law, the magnetic field on the other side of the equation is the total field, including contributions from both enclosed and non-enclosed currents.

Highlights

Coulomb's law is used to calculate electric fields due to point charges, and similarly, Biot-Savart law is used for magnetic fields due to point current elements.

Gauss's law helps calculate electric fields in symmetric situations, and there is an equivalent law in magnetism called Ampere's circuital law.

Ampere's law involves drawing a closed loop in space and calculating the magnetic field at every point along the loop.

The direction of walking around the loop and the orientation of the tiny length vector dl are arbitrary.

Ampere's law states that the integral of the dot product of the magnetic field and dl around a closed loop equals the product of the permeability of free space and the total current enclosed by the loop.

The enclosed current is defined as the total current passing through a surface attached to the loop.

The right-hand rule is used to determine the positive direction of the current relative to the loop.

Ampere's law can be applied to calculate the magnetic field in symmetric situations, simplifying calculations.

Ampere's law and Biot-Savart law are equivalent, and one can be derived from the other.

The magnetic field considered in Ampere's law is due to all currents, both enclosed and non-enclosed by the loop.

The contributions of non-enclosed currents to the magnetic field around the loop often cancel out, simplifying the law's application.

A surface can be any shape attached to the loop, not necessarily flat, to determine the enclosed current.

Attaching a surface to the loop is useful in complex situations where it's not obvious which currents are enclosed.

Ampere's law is particularly useful for calculating magnetic fields around conductors in symmetric configurations.

The law helps in understanding the relationship between electric currents and the magnetic fields they generate.

Transcripts

coulomb's law helps us calculate

electric field due to point charges and

similarly in magnetism bo sawar law

helps us calculate magnetic fields due

to point current elements

but we've also explored gauss's law

which helps us calculate electric fields

in symmetric situations and we've seen

that one can be obtained from the other

they are equivalent

and so now the question is do we have

something similar in magnetism something

that can help us calculate magnetic

fields for symmetric situations the

answer is yes we have something called

ampere's circuital law and in this video

we're going to ask mr ampere to help us

understand his law

so mr amp here what's your law tell us

ampere says let's take an example

imagine we have three

wires that carry some current i1 i2 i3

now ampere says draw a closed loop

and i ask what do you mean he says

anywhere in space any shape you want

draw a closed loop i say okay cool let

me do that so let's say i draw a closed

loop that goes somewhat like

this

i am excluding i3 because

why not he asked us to draw anywhere i

want

fine what next amp here next he says

walk around this loop and i ask well

there are two ways to walk either this

way or that way which way should i walk

he says any direction you want and i

love that i love the freedom that he's

giving us so let me walk this way i'm

walking amp here what should i do now

and now comes the important part he says

at every point

i'll write this at every point

find the dot product of the magnetic

field

and a tiny length dl vector

okay what does that mean

so here's what ampere is saying at every

point in space the three currents are

together producing a magnetic field

right we know that current produces

magnetic field so maybe at this point

just drawing random directions now maybe

the magnetic field is this way

maybe there's a point over here where

the magnetic field is i don't know maybe

this way

and maybe there's a point over here

where the magnetic field is this way

now what amp here says is that as i'm

walking

at every point take a tiny step length

which is dl and we'll have a direction

of this you know tangential to this path

so over here dl will be

this way

over here dl would be this way

and over here because i'm walking like

this dl would be this way you can

imagine dl to be a very tiny step like a

nanometer or something it's an

infinitesimally small step

and

take a dot product of them scalar

product of them so bdl cos theta you

might know how to take the dot product

by now

and do that everywhere and he asks us to

then take a summation of that so add all

of that up over the entire loop and

since we're dealing with infinitesimals

we're dealing with calculus addition in

calculus summation in calculus is what

we call integral

and so this is what ampere wants us to

do in that closed loop take the integral

of b dot d l everywhere now ampere is

warning us not warning sorry reminding

us that this will only work for closed

loops so for example if i had chosen say

a loop which looked like this

then it will not work ampere says don't

take this even here i would take i can

take b dot dl right but ampere says no

no only for closed loops and so to

remind us he's going to put a circle

over here oops let's use the same color

circle over here

and says closed loop all right

all right what what happens if i do that

ampere asks me mahesh what do you think

will happen if you took this uh integral

over the closed loop and i say i have no

idea you tell us amp here this is where

ampere smiles and laughs and says ha the

answer is going to be and this is the

circ this is the ampere circular law the

answer is going to be always

mu naught

times i

enclosed

not available

now before we continue i'm sure a lot of

questions are brewing up in your mind

like why do we need this law well as we

will see in future videos we can use

this to figure out the strength of the

magnetic fields in certain symmetric

situations we'll do that in the future

videos okay

but for now let's concentrate on the

right hand side and ask ourselves what

does this enclosed current mean

well let's ask ampere up here what is

this enclosed current well one way to

think about this is basically how much

current is enclosed the total current

enclosed by the loop but ampere is a

little bit more specific

ampere says look to figure this out

first step you have to do is attach a

surface to this loop and i don't

understand that as ampere what do you

mean so ampere says okay imagine this

imagine you took this and dipped in a

soap solution what would happen there

will be some soap film attached over

here right

here you go let's imagine that's the

soap solution

now ampere says

enclosed current is the current that

punches through this surface whatever is

punching to that surface is the enclosed

current so the enclosed current is

basically the total current that is

passing through

an attached surface to the loop the

attach surface is our soap solution

so in our example what would be the

value of

b dot dl according to ampere's law

well that's going to be mu naught times

what is i n closed only i1 and i2 are

passing through the attached surface

they are the only ones enclosed i3 is

not so i3 will not be in the picture so

the total will be

total will be i1

plus

i2

but the moment i write that i feel

uncomfortable because i know that one

current is going up another current is

going down so one must be positive and

one must be negative right ampere says

yes one must be positive one must be

negative but how do i figure out which

one is positive and which one is

negative what do i do

so ampere says we use the same thing

that we've used so far in magnetism

right hand rule he says take your right

hand and curl it in such a way that the

curved fingers are in the direction of

your travel

and then the thumb represents the

positive direction so in our example

since i'm traveling this way if i take

my right hand

and if i curl my fingers in that

direction my thumb will point downwards

and so this means according to my right

hand thumb rule

downward direction

is positive

for this loop

so i2 would be positive i1 would be

negative

so this

is now the correct application of

ampere's law

for this case

why don't you quickly try one

so let's say let's let's take another

loop which is over here i'm going to

take a rectangular loop because shape

doesn't matter

so let's say we take a rectangular loop

somewhat like this

and this time let's say we walk this way

and calculate b dot dl

can you pause the video and think about

what will be

the closed loop integral of b dot dl

over here is going to be mu naught times

something what will that be can you

pause and think about this

all right so the first step would be to

dip this in a soap solution and attach a

flat surface to it

and now the current that penetrates

through this surface will be our

enclosed surface and you may be

wondering why should we attach a surface

we'll talk a little bit about that

towards the end of the video but the

current that penetrates is i2 and i3

and now we need to know which direction

is positive for that we use our right

hand thumb rule in this case we are

moving in this direction and so if i use

my right hand now

let me keep it over here somewhere okay

if i move the right hand now now notice

the thumb points upwards so upwards is

my

positive so this is now

positive

so what i end up getting is

plus i3 so i3 becomes my positive

current i2 becomes my negative current

so

minus i2

and i1 is not in the picture because i1

is not penetrating to that surface and

there you go this is how we use ampere's

circular law

now before we wind up i want to talk

about some important characteristics of

this law

first of all this law can be derived

from a bo savar law and you can derive

bo sawar law from this law so they're

both equivalent and we use whichever one

is more convenient in our given

situations in some in sometimes when

things are very symmetric we go for

ampere's circular law because it makes

our calculations simpler again something

we'll see in future videos

secondly on the right hand side we only

consider currents that are enclosed by

the loop right so for in this example

only i1 and i2 but not i3 but what about

the magnetic field on the left hand side

is that only due to the enclosed

currents no that is the total magnetic

field so the magnetic field which we are

considering is due to all the currents

enclosed and non-enclosed so how does

that work why is it on one side we have

total field but on the other side only

the enclosed one matters well that's

because again this is like

mathematically we will not get into the

details but what happens is what this

means is that the contribution of bdl

provided by the non-enclosed currents

they add up and become zero so they end

up be giving zero contributions so you

can imagine as you walk around this loop

when you in some cases the contribution

of this is positive uh in some places

the contribution is negative and so the

total contribution of them is zero this

is very similar to what we saw in

gauss's law the char uh you know the

total electric flux only depends upon

the charges that are enclosed by the

surface over there right the charges

which are outside they will contribute

to zero flux same thing very similar

happening over here

finally when it comes to the surface we

said imagine a soap solution attached to

it right but here's the thing about soap

solutions you don't have to have them

flat

if you blow on a soap solution you end

up having an open surface attached to

our loop so the loop becoming the

opening to that surface we can also

attach such surfaces over here so

imagine somebody is blowing from the top

what will happen to that surface we

might get something like this and now

the iron closed becomes the current that

is penetrating through this surface

and so you can attach any open surface

you want to your loop flat being the

simplest one but you will always end up

with the same value of iron closed

now finally finally you may ask why

should we even attach a surface i mean

what's this business of attaching

surface can't i just look at this loop

and tell what is the enclosed current

sure in most simple situations yes

but in general we can have very complex

situations in which it may not be so

obvious and i'll not dig too much into

it we will look at one such situation

sometime in the future where we deal

with calculating the magnetic field when

there is a capacitor things will become

much more interesting over there and

this attachment of surface will make a

lot more sense over there

but for now it's completely fine in most

of our examples it's fine if you don't

attach it but ampere suggests you attach

a surface to our loop and find the

current that is punching through that

surface that becomes our ion closed