Uploaded by cfurse on 01.10.2009

Transcript:

Welcome to ECE 3300 at the University of Utah. In Lecture No. 16,

we're going to talk about the Biot-Savart Law. The Biot-Savart

Law is to the magnetic field like Coulomb's Law is to the electric

field. And that's why we're going to study it right now together.

The operations we do, the calculations, for the Biot-Savart Law

and Coulomb's Law are very, very similar, and I'll be showing you

what I mean. Let's consider kind of a grid of possibilities.

Let's talk about the electric field up here on the top. And let's

talk about the magnetic field down on the bottom. If we have an

electric field distribution or charged distribution that is just

kind of random, maybe there are some point charges, maybe there

are a few line charges, then we apply Coulomb's Law. If we have a

nice symmetric charged distribution, and perhaps we want to find

the electric field out here, then we can apply Gauss's law for the

electric field. You'll learn about Gauss's law later. Now,

remember I said that the Biot-Savart is to the magnetic field like

Coulomb's Law is to the electric field. So down here, if I have a

random current distribution, maybe I have a combination of line

currents, maybe I have a few point currents, maybe I have some

surfaces, then I can use the Biot-Savart Law. But if I have a

nice, uniform, symmetric current distribution, then I can apply

Faraday's Law in order to find the magnetic field at a point out

here. So Coulomb's Law and Biot-Savart Law can be used for random

distributions of charges in the case of coulombs; current in the

case of Biot-Savart, Gauss's Law for the electric field, and

Faraday's Law only apply when we have symmetric charges in the

case of Gauss's law and currents in the case of Faraday's Law.

This is the Biot-Savart law. It says that the magnetic field here

is equal to the current, crossed with a vector, divided by 4 pi

times the vector squared. If we have a current going in this

direction I, so perhaps this is an I-Z current, and we want to

find the magnetic field right out here, we define a unit vector R,

which is in the direction between our source and our field point.

The magnitude of the distance between the source and the location

of H is R. So I take my I and I cross it with this unit vector,

and that gives me the top portion of the Biot-Savart. I find the

distance between my source and my magnetic field and that gives me

the distance R squared. So in this lecture, we're going to learn

how to use the Biot-Savart Law to find H or B from a current

distribution. Those are vectors. The current distributions could

be line, surface or volume in three different coordinate systems.

So let's first consider the different types of current

distributions I might have. If I had a simple current going in

one direction, I, then I could use a line surface representation.

I'm going to change my notation a little bit from the book because

I want you to notice that the vector that's represented here is

actually the direction that the current is going, not specifically

the direction that the integration variables are going. So here's

my current and it's going in the Z direction. If I want to find

the magnetic field out here, I'm going to define my unit vector in

the direction from my source to the location where I want to find

the field. If I had something like a strip, a surface, I might

have a surface current distribution JS. JS would be given in amps

per meter squared, and notice it does have a direction. A third

possibility might be something like a cylinder, a volume of

current, and that would be a JV. Notice that the current again

has a direction and this is a volume current density given in amps

per meter cubed. So these are the three different current

distributions that I may use in Biot-Savart: Line, surface and

volume. And we'll be looking at each one of those separately.

we're going to talk about the Biot-Savart Law. The Biot-Savart

Law is to the magnetic field like Coulomb's Law is to the electric

field. And that's why we're going to study it right now together.

The operations we do, the calculations, for the Biot-Savart Law

and Coulomb's Law are very, very similar, and I'll be showing you

what I mean. Let's consider kind of a grid of possibilities.

Let's talk about the electric field up here on the top. And let's

talk about the magnetic field down on the bottom. If we have an

electric field distribution or charged distribution that is just

kind of random, maybe there are some point charges, maybe there

are a few line charges, then we apply Coulomb's Law. If we have a

nice symmetric charged distribution, and perhaps we want to find

the electric field out here, then we can apply Gauss's law for the

electric field. You'll learn about Gauss's law later. Now,

remember I said that the Biot-Savart is to the magnetic field like

Coulomb's Law is to the electric field. So down here, if I have a

random current distribution, maybe I have a combination of line

currents, maybe I have a few point currents, maybe I have some

surfaces, then I can use the Biot-Savart Law. But if I have a

nice, uniform, symmetric current distribution, then I can apply

Faraday's Law in order to find the magnetic field at a point out

here. So Coulomb's Law and Biot-Savart Law can be used for random

distributions of charges in the case of coulombs; current in the

case of Biot-Savart, Gauss's Law for the electric field, and

Faraday's Law only apply when we have symmetric charges in the

case of Gauss's law and currents in the case of Faraday's Law.

This is the Biot-Savart law. It says that the magnetic field here

is equal to the current, crossed with a vector, divided by 4 pi

times the vector squared. If we have a current going in this

direction I, so perhaps this is an I-Z current, and we want to

find the magnetic field right out here, we define a unit vector R,

which is in the direction between our source and our field point.

The magnitude of the distance between the source and the location

of H is R. So I take my I and I cross it with this unit vector,

and that gives me the top portion of the Biot-Savart. I find the

distance between my source and my magnetic field and that gives me

the distance R squared. So in this lecture, we're going to learn

how to use the Biot-Savart Law to find H or B from a current

distribution. Those are vectors. The current distributions could

be line, surface or volume in three different coordinate systems.

So let's first consider the different types of current

distributions I might have. If I had a simple current going in

one direction, I, then I could use a line surface representation.

I'm going to change my notation a little bit from the book because

I want you to notice that the vector that's represented here is

actually the direction that the current is going, not specifically

the direction that the integration variables are going. So here's

my current and it's going in the Z direction. If I want to find

the magnetic field out here, I'm going to define my unit vector in

the direction from my source to the location where I want to find

the field. If I had something like a strip, a surface, I might

have a surface current distribution JS. JS would be given in amps

per meter squared, and notice it does have a direction. A third

possibility might be something like a cylinder, a volume of

current, and that would be a JV. Notice that the current again

has a direction and this is a volume current density given in amps

per meter cubed. So these are the three different current

distributions that I may use in Biot-Savart: Line, surface and

volume. And we'll be looking at each one of those separately.