Biot Savart law (vector form) | Moving charges & magnetism | Khan Academy

Summary

This video explores how to calculate the magnetic field produced by a current-carrying wire, focusing on the Biot-Savart Law. The law involves analyzing tiny segments of the wire and using vector equations to determine the magnetic field's strength at a given point. The video explains the relationship between current, distance, angle, and the magnetic field. It also introduces constants like permeability and demonstrates techniques like the right-hand rule and cross products to find the magnetic field's direction. Ultimately, it highlights how magnetic fields are calculated through integration for entire wires.

Takeaways

• 🔋 Current-carrying wires generate a magnetic field around them, calculable via the Biot-Savart law.
• 🧲 To apply the Biot-Savart law, you must focus on a small segment of the wire, not the entire length.
• 📏 The law involves calculating the magnetic field due to a tiny wire segment of length 'dl' at a distance 'r' from the segment.
• ✖️ The magnetic field, represented by 'dB', is a vector and calculated using a cross-product involving 'dl' and 'r'.
• 📐 The magnitude of the magnetic field depends on current (I), the length of the current element (dl), and the sine of the angle between dl and r.
• 🌀 Magnetic field strength decreases as the distance from the current element increases, following an inverse-square law (1/r²).
• 📏 The magnetic field is strongest at 90 degrees relative to the current element and weakest (or zero) along the axis (0 or 180 degrees).
• 🧲 The constant μ₀ (permeability of free space) is a fundamental value, approximately 4π × 10⁻⁷ Tesla meters per Ampere.
• 👋 The direction of the magnetic field can be determined using the right-hand clasp rule or by using the vector cross-product.
• 🔄 To calculate the total magnetic field of the entire wire, sum the contributions of all small segments (integral).

Q & A

What is the Biot-Savart law, and why is it important in calculating magnetic fields?

The Biot-Savart law is a fundamental equation in magnetism that describes the magnetic field produced by a current-carrying element of wire. It is important because it allows the calculation of the magnetic field's strength and direction at any point in space around the wire.

Why can't we calculate the magnetic field for the entire wire at once using the Biot-Savart law?

The Biot-Savart law requires dividing the wire into infinitesimally small elements (dl) to calculate the magnetic field, as it only applies to small current elements. Summing the magnetic field contributions of each tiny element gives the total magnetic field around the entire wire.

What does the vector form of the Biot-Savart law tell us about the magnetic field?

The vector form of the Biot-Savart law shows that the magnetic field (dB) is a vector that depends on the current (I), the length of the wire element (dl), the unit vector in the direction of the point of interest (r̂), and the distance from the current element (r). It also includes a cross product, which means the magnetic field depends on the angle between the current element and the position vector.

How does the magnitude of the magnetic field vary with distance from the wire?

The magnitude of the magnetic field decreases with the square of the distance from the wire, following the inverse-square law (1/r²). This means that the farther you are from the wire, the weaker the magnetic field becomes.

What role does the angle (θ) play in the Biot-Savart law?

The angle θ is the angle between the direction of the current element (dl) and the position vector (r̂). The magnetic field is proportional to the sine of this angle (sinθ), meaning the magnetic field is strongest when θ = 90° and zero when θ = 0° or 180°.

What is the significance of the constant μ₀ in the Biot-Savart law?

The constant μ₀, called the permeability of free space, determines the strength of the magnetic interaction in a vacuum. Its value is 4π × 10⁻⁷ T·m/A, and it plays a similar role to the permittivity constant in Coulomb's law for electric fields.

How do you determine the direction of the magnetic field produced by a current element?

The direction of the magnetic field is determined using the right-hand rule. Point your thumb in the direction of the current, and your fingers will curl in the direction of the magnetic field around the wire. Alternatively, the direction can be found using the cross product of dl and r̂.

Why does the magnetic field differ from that of an electric field in terms of directionality?

Unlike electric fields, which radiate uniformly from a point charge, the magnetic field produced by a current-carrying wire is directional due to the cross product between the current element and the position vector. This creates a magnetic field that is strongest perpendicular to the current and zero along the current's axis.

What happens to the magnetic field at different points around the current element?

At points perpendicular to the current element, the magnetic field is at its maximum because the angle between the current and the position vector is 90°, making sinθ = 1. At points along the current’s axis, the magnetic field is zero since the angle is 0° or 180°, making sinθ = 0.

How can you calculate the total magnetic field around a wire using the Biot-Savart law?

To calculate the total magnetic field around a wire, you must integrate the magnetic field contributions (dB) from each infinitesimal current element (dl) over the entire length of the wire. This integration accounts for all the small magnetic fields generated by each segment of the wire.

Outlines

00:00

🔍 Introduction to Biot-Savart Law

This paragraph introduces the concept of the Biot-Savart Law, which describes how to calculate the strength of the magnetic field produced by a current-carrying wire. The law focuses on a tiny element of the wire, represented by its infinitesimal length dl. The formula for the magnetic field (dB) involves a cross product between the wire element and the position vector, scaled by constants. The explanation emphasizes the vector nature of the equation, how it parallels Coulomb’s Law, and the significance of each component, such as the current I, distance r, and angle between vectors.

05:02

🧭 Comparing Magnetic and Electric Fields

This section compares the electric field around a point charge and the magnetic field around a current element. The electric field is uniform at a fixed distance from a charge, but the magnetic field varies due to the sine of the angle between the current element and the position vector. The discussion shows how the angle affects magnetic field strength: it's maximum when perpendicular (90 degrees) and zero when aligned (0 or 180 degrees). This highlights a critical difference between electric and magnetic fields and emphasizes that magnetic fields are directional and dependent on angle.

10:03

🌀 Direction of Magnetic Fields and Cross Product

This paragraph delves into finding the direction of the magnetic field produced by a current element. Two methods are discussed: the right-hand clasp rule and the vector cross product. The clasp rule involves pointing the thumb in the direction of the current, with the encircling fingers indicating the field direction. The cross product method uses aligning and crossing vectors dl and r with the right hand to determine the direction of the magnetic field. Examples show the consistency of both methods in predicting field directions at various points around a current element. The conclusion highlights that integrating these tiny magnetic fields can calculate the total field produced by an entire wire.

Keywords

💡Biot-Savart Law

The Biot-Savart Law describes how to calculate the magnetic field produced by a small segment of current-carrying wire. It is a fundamental principle in electromagnetism, stating that the magnetic field is proportional to the current and inversely proportional to the square of the distance from the current element. The video introduces this law as the method to calculate the magnetic field strength around a wire and uses it to explain how small current elements contribute to the overall magnetic field.

💡Current Element

A current element is a very small segment of a current-carrying wire, denoted by 'dl' in the video. The Biot-Savart Law applies to each of these tiny elements, and the magnetic field generated by each one is calculated and then summed to find the total magnetic field. In the script, the speaker refers to these small segments multiple times to emphasize the importance of considering small portions of the wire when calculating magnetic fields.

💡Magnetic Field

The magnetic field is the invisible force field created around current-carrying conductors. The strength and direction of the magnetic field are determined by factors such as the current, distance, and angle with respect to the current element. The video explains how the Biot-Savart Law helps calculate the strength of the magnetic field at different points around the wire, illustrating its dependence on distance and angle.

💡Right-hand Rule

The right-hand rule is a mnemonic for determining the direction of the magnetic field relative to the current's direction. In the video, the presenter explains that if you clasp a conductor with your right hand so that your thumb points in the direction of the current, your fingers will curl in the direction of the magnetic field. This rule helps visualize the orientation of the magnetic field lines around a current-carrying wire.

💡Cross Product

In vector mathematics, the cross product determines the direction of the magnetic field in the Biot-Savart Law. The video introduces the cross product between 'dl' (current element) and 'r' (the distance vector), and explains that it incorporates both magnitude and angle (theta). The sine of the angle between these vectors affects the magnetic field’s magnitude, which is a key difference between magnetic fields and electric fields.

💡Sine Theta (Sin θ)

Sine theta refers to the sine of the angle between the current element and the point where the magnetic field is measured. It affects the magnitude of the magnetic field, making it maximum when the angle is 90° and minimum (zero) when the angle is 0° or 180°. The video emphasizes this as a crucial factor in understanding how the magnetic field varies at different points around the wire.

💡Permeability of Free Space (μ₀)

The permeability of free space (μ₀) is a constant that describes how much magnetic field is produced in a vacuum by a given current. It has a value of 4π × 10⁻⁷ tesla meters per ampere. The video introduces this constant when discussing the Biot-Savart Law, explaining that it acts similarly to how permittivity governs electric fields in Coulomb’s law.

💡Inverse Square Law

The inverse square law is a principle that describes how the strength of a force or field diminishes as the square of the distance from the source increases. The video highlights that the magnetic field generated by the current element follows this rule, meaning it decreases as 1/r² as the distance (r) from the current element increases. This mirrors the behavior of electric fields and gravitational forces.

💡Unit Vector (r̂)

A unit vector, denoted as 'r̂' in the video, is a vector that indicates direction without affecting magnitude. It points from the current element to the point where the magnetic field is being calculated. The video explains that since 'r̂' has a magnitude of 1, it only influences the direction of the magnetic field, while its length does not contribute to the magnitude.

💡Clasp Rule

The clasp rule is another name for the right-hand rule used to determine the magnetic field direction. The video demonstrates how clasping your hand around the conductor, with your thumb pointing in the direction of the current, gives the direction of the magnetic field as your fingers curl around the wire. This rule complements the vector approach to understanding magnetic fields and helps visualize them.

Highlights

The Biot-Savart law is a famous law in magnetism that helps calculate the strength of the magnetic field produced by a current-carrying wire.

The law requires considering a tiny element of the wire, not the entire wire, to calculate the magnetic field.

The magnetic field at a point due to a tiny wire segment is given by the equation: μ₀/4π * (I * dL × r̂) / r².

The magnitude of the magnetic field is proportional to the current (I) and the tiny element length (dL), and inversely proportional to the square of the distance (r²).

The sine of the angle (θ) between the direction of the current (dL) and the distance vector (r) plays a crucial role in determining the magnitude of the magnetic field.

The magnetic field is stronger when the distance from the current element is smaller and the angle between the current direction and the distance vector is 90 degrees.

The Biot-Savart law shows that the magnetic field depends not only on the distance but also on the angle between the current direction and the position vector.

At 90 degrees, the magnetic field is at its maximum, while at 0 or 180 degrees, the magnetic field is zero.

The constant μ₀ (permeability of free space) has a value of 4π × 10⁻⁷ T·m/A and plays a key role in the Biot-Savart law.

Magnetic fields are always maximum perpendicular to the current element and minimum along its axis.

To find the direction of the magnetic field, the right-hand clasp rule is used, where the thumb points in the direction of the current, and the encircling fingers give the direction of the magnetic field.

The direction of the magnetic field can also be found using the cross product between the current element and the distance vector.

At any point, the magnetic field direction can be determined using either the right-hand rule or by calculating the cross product.

For a tiny element of wire, integrating the magnetic field contributions from each piece yields the total magnetic field for the entire wire.

Magnetic field strength behaves similarly to the inverse square law seen in Coulomb's law and Newton's law of gravity.

Transcripts

we know current carrying wires produce

magnetic field around them in this video

let's figure out exactly how to

calculate the strength of that magnetic

field and it's given by a famous law in

magnetism which is called the bo savar

law that's how you pronounce them

they're frenchmen

so

what does the bosavar law say

for to use this law we have to consider

a tiny element of the wire you can't

just do the

you can't just calculate it for the

entire wire so let's say i consider a

very tiny piece of that wire

all right that very very tiny piece of

that wire and let's say that that piece

has

a length of dl

that's the length of that and i'm using

d because it's very tiny you imagine

it's an infinitesimal

and let's say that the current

is i

and now imagine i want to calculate what

the strength of that magnetic field is

at some point over here some random

point over here

at distance r

from this particular element so how do i

calculate the strength of the magnetic

field here so b or saw our law basically

says that the strength of the magnetic

field at this point and we'll call that

d b

b stands for magnetic field and d

because it's a tiny magnetic field

created by a tiny piece of wire

so that's going to be vectorially so

it's a vector equation get ready for

this

it's going to be a constant

mu naught by 4 pi

times i

times d l

cross

r cap

divided by

r squared

now i always found vector equations a

little scary because you know they have

all these cross products and everything

so first thing we'll do is look at a

magnitude of this equation and try to

make sense of that so what would be the

magnitude of this equation so if i just

look at the magnitude of db which i'm

just going to write db that's going to

be this is a constant we'll get to that

we'll get to what that mu naught is so

mu naught by 4 pi

i will remain the same

denominator r square will remain as it

is what is the magnitude of this how do

you take magnitude of a cross b it will

be magnitude of a

times magnitude of b

times sine of the angle between the two

so magnitude of this that's just going

to be dl

times magnitude of this what is this by

the way what is this r cap our cap is a

unit vector in the direction of r

and since it's a unit vector its

magnitude is one so magnitude of this

times magnitude of this which is 1 times

the sine of the angle between them so

sine of the angle between the dl vector

how what is the direction of dl vector

well you choose the direction of the

current as the direction of the dl

vector

and you choose the direction of r as a

direction of r cap so

the angle between the two is going to be

this if i call this theta

then it will be sine

theta because the cross product has a

sine in it so sine theta

so that's the magnitude

all right so what is the equation saying

well

first of all there is a constant over

here just like how we have constants in

coulomb's law so there we had 1 over 4

pi epsilon naught here we have mu naught

by 4 pi i'll get to that mu naught part

in a second okay but there is a constant

and then it says that magnetic field

depends upon the strength of the current

which makes sense because we know it's

the moving charges that create magnetic

field so higher current means more

moving charges per second more magnetic

field that makes sense

we also see that the db depends upon dl

the length of the current element why is

that

well that's because if you took a longer

current element then you would have more

moving charges in them and as then so

the magnetic field will be due to more

moving charges and so you would expect

the magnetic field to increase so if you

have double the dl you have double the

number of moving charges and so you have

double the magnetic field so this also

makes perfect sense

what does this say oh it's inversely

proportional to r square that means that

if you go farther away the magnetic

field drops off as one over r square and

that actually makes me feel very

comfortable because we've seen this

before in newton's law of gravity

we've seen one over r squared in

coulomb's law and now we're also seeing

this in bosavar law so that's that's

great and just like with coulomb's law

how it only works for point charges

we're seeing that bo sub r only works

for point current elements so we have to

only consider very tiny pieces of wire

you can't consider that for the entire

length of wire so

so far you know everything was very

similar to coulomb's law but there's one

major difference there's a sine theta

coming over here and that's a huge

difference

and let me show you how that difference

you know actually pans out

so if we go back to our charges and

electric fields

imagine i consider a circle around a

point charge which is at the center and

i asked you hey consider three points a

b and c where do you think the electric

field not magnetic electric field

strength would be higher where do you

think would it be the charge is at the

center where would it be

well

because the distance is the same it

doesn't matter everywhere electric field

on this circle is going to be the same 1

by 4 pi

epsilon not q by r square q is the same

r is the same everywhere okay but now

let's consider replace this charge with

a current element

okay a tiny piece of wire having current

is called current element

now i ask you again um

where do you think would the magnetic

field be higher would it be same

everywhere would be different can you

pause the video just look at this sine

theta

and see if you can figure out out of

three places where it would be higher

will be lower

yeah there would be can you pause and

try

all right

so this time we need to be a little bit

more careful because we also have to

consider the angle theta between the

current and the r so the current is to

the right what is the direction what

what's where is r over here for this one

our r is going to be from here to here

okay

here to here

and so over here

this is 90 degrees

sine 90 is 1 that's the maximum value

meaning over here you will get the

maximum value of

db

magnetic field okay what about at this

point

well if i were to draw from here to here

the r value remains the same

but look at what happens to theta it's

no longer 90 degrees it's more than 90.

and sine of any angle more than 90

between 90 and some obtuse angle it's

going to be less than one so db is going

to be smaller

what about over here

well if i were to draw again

uh r from here to here

if i do that

this time what's the angle the angle is

zero

what is sine zero oh sine zero is zero

so over here

our db

is zero

look at what what you're seeing

even though we're going at the same

distance the magnetic field is not

exactly the same because it not only

depends on the distance it also depends

on the angle that's the most important

thing over here

we see that when you are at 90 degrees

whether you are here or you are here

somewhere 90 degrees you get maximum

field

and what we see is that at 0 or at 8 180

degrees you know if you go here this

will be a 180 degrees sine wave is also

zero you get zero

and so in between

you get magnetic field in between the

maximum and the minimum value so it

decreases and then it increases and then

again decreases and then it increases

again

so magnetic fields will always be

maximum

perpendicular to the current element and

they'll be minimum on the axis

of that current element and that's what

this this important thing is telling us

all right now final couple of things one

is what is this mu knot what's important

is it's a constant

for vacuum and most of the times we'll

be dealing with vacuum and the value of

that constant is going to be let me

write that down over here the value of

that constant is 4 pi times 10 to the

power

-7 and it'll have some units which you

can work out this is tesla and you have

current i don't remember the units i

think you can work that out

but it's given a name it's called per

me a oops you can't i can't read this

sorry

per

me a b t

of vacuum

and if you think hey that sounds very

familiar to what we saw earlier yeah

that was called permittivity this is

called permeability i did not name it

don't blame me i know these names are

very very similar to each other what

matters to me is that hey i know the

value of that and it's it's 4 pi times

10 to the power minus 7. and so

basically when you look at this whole

constant the value of the constant is

just 10 power minus n because 4 pi would

just cancel out

all right

last thing let's look at the direction

of the magnetic field because that's

also important magnetic field is a

vector it has a direction how do i

figure out the direction of the magnetic

field there are a couple of ways to do

that i like both of them one is

something that we've already seen before

to find the direction of the magnetic

field we can use our right hand clasp

rule so you take your right hand

and you clasp the conductor so that your

thumb points in the direction of the

current then the encircling fingers will

give you the direction of

the magnetic field and then you can use

that to figure out what the magnetic

field direction would be so everywhere

to the right doesn't matter where you go

everywhere to the right the magnetic

field is into the screen so immediately

i can say hey the magnetic field at

point p should also be into the screen

the magnetic field somewhere to the left

would be out of the screen

but

we have a vector equation we should also

be able to get the same answer just by

looking at this vector equation so let's

try that let's get rid of this class

rule you can always use the class rule

but you know using two two methods are

always better you can always check

yourself so over here uh if you want to

get the direction of magnetic field you

have to get the direction of dl cross r

so here's how i like to do i have look

at my dl

it's this way i look at my r is this way

so i have to do across from dl to r

and how do you do cross from dl to r you

take your right hand and you align it

such that your four fingers are along

this cross here's how i would do it so

if i would show you my hand i would

align it with my dl

and then i cross it this way

and while i do that look at the

direction of my thumb the thumb gives

you the direction of the cross product

the thumb is pointing inwards and

therefore the magnetic field over here

must be inwards so both methods the

clasp rule and the cross product will

give you this

same answer so can you quickly find out

what will be the direction of the

magnetic field at point a and point b

can you pause and find that out

all right if you use the clasp rule then

we clasp our conductor

such that the thumb points in the

direction of the current

that gives you the magnetic field

encircling fingers gives the magnetic

field i see that at point a the magnetic

field is coming out of the screen

so immediately i understand that the

magnetic field over here must be

out of the screen and we show out of the

screen this way

and over here

everywhere down the magnetic field

should be into the screen so everywhere

below it should be into the screen

but can we also confirm that using our

cross product of course

so if i start with say at point a

i have to cross from dl

to r so i have to cross this way so my

encircling finger should go this way and

so the way i align my palm is like this

preparing it to cross

and when i cross it with my circle my

four fingers

that's how it looks and so that look at

the direction of the thumb it's coming

out of the screen that's exactly what we

predicted

and similarly if i were to do at point b

this time i have to cross it the other

way around and so i'm going to hold my

palm the other way around like this

if i hold it like this now i cross it

in this direction and so my thumb

represents the magnetic field is into

the screen it's exactly what we get over

here

and so now that we know how to calculate

magnetic field due to tiny pieces of

wire if you want to calculate the total

magnetic field due to all the to the

entire wire we just sum them up due to

each tiny piece or we have to do an

integral and we'll look at some problems

in future videos