ECE3300 Lecture 16-1 Biot Savart Law overview

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.