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What do rain, bees, and vectors have in common?
They're all ways that MIT physicists visualize
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the concept of "flux." In this video, we'll
explore flux as it relates to Gauss' law.
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If you've ever had trouble choosing the right
Gaussian surface, getting a quantity out of
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an integral, or just wanted to know what flux
is, get ready: this video is for you.
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This video is part of the Derivatives and
Integrals video series.
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Derivatives and integrals are used to analyze
the properties of a system. Derivatives describe
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local properties of systems, and integrals
quantify their cumulative properties.
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"Hello. My name is Peter Fisher. I am a professor
in the physics department at MIT, and today
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I'll be talking with you about electric flux
and Gauss' Law.
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By this time in your course you've seen and
used Gauss' Law quite a bit. You have a good
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handle on the electric field, what it does
and how it works. We will also assume that
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you know how to take an area integral, a skill
that is important to Gauss' Law but which
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is not taught in this video.
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Our end goal is to improve your ability to
use Gauss' Law. To do this, we'll help you
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develop an organized view of electric flux
and what goes into it. We'll also spend a
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lot of time talking about symmetry and how
to use it to your advantage.
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We're going to start by thinking about what
goes into flux and the interrelationships
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between these quantities, so that we can use
flux more effectively.
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As a refresher, here are the two equations
we'll be working with today: the definition
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of flux and Gauss' Law. You may have seen
these written with a double integral, or with
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slightly different notation, so take a quick
look to make sure you're familiar with all
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the pieces.
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To really use symmetry to our best advantage
in Gauss' Law, we need to understand a few
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more basic things. We need to be able to use
vector areas, to categorize charge distributions,
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and to determine when and how the two can
work together.
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The underlying reason we want to use symmetry
is that integrals are difficult, significantly
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more difficult than derivatives. There are
many situations in which we cannot determine
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the flux integral. Therefore, we seek to simplify.
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The integrals we're dealing with are surface
integrals, also called area integrals. There
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are two basic kinds, open and closed, as you
can see on the screen. An open surface uses
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an open integral, and a closed surface uses
a closed integral. Flux can be defined for
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any surface, but Gauss' Law uses only closed
integrals.
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When we take the integral, we use dA, a tiny
piece of the area. dA is a vector piece of
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the area, so it has a direction to it as well.
dA always points perpendicular to the surface.
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With closed surfaces we usually choose dA
to point outward from the surface.
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dA usually gets represented as two other differentials,
such as dx times dy. Which two depends on
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the coordinate system you choose, with some
being more complex than others. Sometimes
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we will need these, so it's good to have them.
The good part is that if we use symmetry correctly,
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we can avoid having to do an integral at all.
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Let's pursue our goal of not doing an integral.
We have this integral of E dot dA. Let's write
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the dot product as a cosine. Under certain
circumstances, we can remove the E from the
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integral. We can only do that when our electric
field has uniform magnitude over the entire
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surface. You can see some examples on the
bottom of the screen.
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We can remove the cosine(theta) from our integral
only when the angle between the area and the
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electric field doesn't change. The left-hand
example shows a field that always points in
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the same direction, and an area that always
points in the same direction. The middle example
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has a field whose direction changes, but it
always points outward, just like the area
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vector from the cylinder points outward. It
still works. In the right-hand example, the
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field has a constant direction, but the area
vector does not. The angle between them changes,
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so we couldn't remove the cosine from the
integral.
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If we want to move both the field and the
cosine out of the integral, we need to fulfill
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both conditions. The left-hand and center
examples only fill one condition each. The
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example on the right has both, and would be
ideal to use.
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Before we move on, a reminder about flux and
angles. Only electric field lines that actually
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pass through a surface provide flux. When
the surface is parallel to the field, no flux
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is provided. This is particularly useful for
some three-dimensional surfaces, such as the
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empty cylinder to the right. If we remember
this, we can really simplify our integrals.
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Another thing to remember: Flux is a scalar.
As long as the field strength is the same,
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and the angle between the field and surface
stays the same, the flux will be the same.
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We don't have to worry about the flux pointing
in a particular direction, because it's just
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a number with no direction.
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Now we know how we would like the field and
the area to line up. Next we need to understand
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the circumstances that make that possible.
We need to look at electric charge distributions
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and their symmetry.
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Electric charge can come as a single point,
as in an electron or proton, but it can also
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take other shapes. It might be stretched into
a line, spread throughout a volume, or spread
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over a surface. Not a Gaussian surface - remember,
that's something we make up to solve a problem.
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Charge is spread on a physical object. You
can stretch the charge out into lines, spread
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it on thick wires or spherical shells, build
a solid ball out of it, or even just smear
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it around on an object.
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An object has symmetry when it has exactly
similar parts that are either facing each
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other, or arranged around an axis. On the
screen are some pictures of objects that have
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symmetry.
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There are three categories that we'll investigate,
and three types of symmetry to consider. Our
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hope is to have a highly symmetric charge
distribution, with any of the three types
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of symmetry. That's when Gauss' Law is easiest
to apply.
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These examples have little or no symmetry
to them, and are not the sort of thing for
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which we can use Gauss' Law.
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These three charge distributions have visible
symmetry, but it's not enough. We need a charge
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distribution so symmetric that the field it
generates is also symmetric.
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Let us show why. Let's say we have a charged
cube, which is certainly a symmetric object.
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We can draw the field from it fairly easily.
If we place a cubical Gaussian surface around
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the charge, we can see the problem: the field
and the surface have different angles in different
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places. There's some symmetry here, but not
enough to allow us to pull everything out
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of the integral.
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These cases are ideal: a point charge, a spherical
shell, an infinite sheet of charge, or an
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infinite cylindrical shell of charge. No doubt
you've used Gauss' Law with these charge distributions
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before. How can we describe such distributions
and find others like them?
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All three of those have a special quality:
if you move in two directions, they look exactly
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the same. No matter how far you move across
the plane in the x or y directions, it looks
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identical. Move along the cylinder or around
it, and it looks the same. Rotate around the
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sphere around the equator or the poles, and
you can't tell the difference.
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We can also add similar objects together or
scale them however we like. By integrating
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the fields from a lot of shells, we can create
a solid object, just like these Russian nesting
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dolls.
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Of course, perfectly symmetric charge distributions
are rare in real life. This is probably because
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infinitely long cylinders are so hard to find.
Thus we often use Gauss' Law in cases where
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the charge distributions are close to one
of our symmetric cases. We can usually obtain
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excellent estimates of the true electric field
in this way.
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Two approximations are particularly common:
when the object is so large that we can treat
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it as being infinite, or when we are so close
to the object that its large-scale features
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become unimportant. In these cases, we often
treat the object as being an infinite plane.
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Here's an example of the first approximation.
Compare the distance from your point of interest
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to the object, to the size of the object itself.
If the smallest dimension of the object is
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ten times your distance from it, you can rely
on a very good estimation from Gauss' Law,
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even if your plane isn't really infinite.
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To use the other approximation, we want a
situation where the curvature is very small
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in the place we care about. For example, it
would be very difficult for us to find the
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field at this point. However, if we zoom in,
we can reach a point where the surface looks
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like a flat plane. Finding an estimate for
the field at this point will be much easier.
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Now we have the background we need. It's time
to put all the pieces together and pick our
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Gaussian surface.
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We'll do a quick review in case you missed
something, and give you an example or two.
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After that, it's your turn to pick the best
surface for a given situation.
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In order to simplify our integrals as much
as possible, we seek out situations with as
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much symmetry as we can get. The symmetry
in the charge distribution isn't something
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we can control, but our choice of surface
will determine whether there is symmetry in
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how the electric field meets the Gaussian
surface. And that's always the key: you choose
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the surface to make the problem easier for
you. Finally, it's important to remember that
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in some cases, Gauss' Law won't be helpful,
and it's good to look for another method.
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Here are three charge distributions: a long
cylinder, a sphere, and a flat plane. To maximize
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the effects of symmetry, we choose surfaces
that match up well with the charge distributions.
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To find the field near a cylinder, we surround
it with another cylinder. To find the field
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near a sphere, we encase it in a sphere. To
find the field near a charged plane, we use
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a box-shaped surface. Sometimes people will
use a cylindrical surface instead, and count
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the flux that goes through the top of the
cylinder. Both approaches are valid.
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Here is your first challenge: choose the Gaussian
surface that will best fit this charge distribution.
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We want to know the electric field at a certain
distance from the center of this solid sphere.
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What surface would you choose?
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A sphere inside our existing sphere will be
easiest. This example helps us remember that
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the Gaussian surface can be inside an object.
The area of the surface will be the usual
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four thirds pi r cubed. We'll need to remember
to use only the charge inside our volume,
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not the whole charge of the sphere.
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Here's challenge number two. We would like
to know the electric charge contained in a
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thundercloud. Let's say that the inside of
the cloud has an electric field that looks
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like this.
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What surface would you choose?
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A box-shaped surface will take advantage of
the Cartesian symmetry in this electric field.
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Because of the alignment between the sides
of the box and the electric field, they will
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not contribute to the flux integral. We can
use just the area of the top and bottom of
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the box, and the field at those locations,
to find the flux. This is a common technique.
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Here's challenge number three: finding the
electric field at a certain distance from
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the center of charged disc.
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What surface would you choose?
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No surface will work! The problem is that
there's not enough symmetry in this situation
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to pick an appropriate surface.
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If we can't get an exact solution, we would
like to approximate the field. Unfortunately,
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if we look at the distances involved, we are
not close enough to the disc to make a good
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approximation.
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In the end, this is not a good Gauss' Law
problem. It is solved much more easily through
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other techniques, such as integration and
Coulomb's Law. An example can be found in
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the supplemental materials for this video.
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Here is the final challenge. A top and side
view of the field are shown.
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What surface would align well with this field?
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A cylindrical surface would work very well
with this circular field. You can see that
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the field is perpendicular to the area vector
at all points. Our dot product is now giving
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us a cosine of ninety degrees at all times.
This means a zero value for the flux at all
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points. We don't even need to determine the
area of the cylinder.
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If our flux is zero, the total charge inside
must also be zero. We might guess that positive
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and negative charges together could create
this field, but as it turns out, they cannot.
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Fields that look like this are created by
a changing magnetic field, using Faraday's
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Law, which you will learn more about later
in your course.
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There are some times when you'll just have
to integrate; there's no way around it. Not
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every situation is perfectly symmetric. When
you come to these situations, look for the
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following things to help you out. First, an
area vector that you can define easily. Second,
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an electric field that you can easily find
or that is defined for you in a problem. Third,
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see if you can create a surface where either
the magnitude of the field or its angle is
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uniform. The more of these you can take advantage
of, the better.
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Let's review.
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The more symmetry we can take advantage of
in our problem, the easier time we'll have
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with Gauss' Law and flux. Ideally, we want
a situation where the charge, the field it
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creates, and the Gaussian surface we choose
all have the same symmetry, so that we can
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simplify the integral as much as possible.
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For the final segment of this video, we wanted
to answer a common question about electromagnetism.
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What is flux? What does it mean? How can we
understand it?
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We took a video camera into the physics department
at MIT to find out how physics professors
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and students think about flux. Here's what
we found out.
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Thank you
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for watching this video. We
hope that it has
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improved your understanding of flux and Gauss'
Law. Good luck in your future exploration
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of electromagnetism.