Current Density and the Continuity Equation Video Lecture - Electrical Engineering (EE)

okay in this video I'm going to continue
on with my tutorials on magneto statics
this is video number two and I'm going
to discuss the continuity equation the
previous video to this which is balanced
is number one
we're discuss the Lorentz force law so
first of all just to remind us we know
that all magnetic phenomena are a result
of the movement of electric charge that
means that there are no magnetic charges
or what we call magnetic monopoles this
particular video I'm going to discuss
current density I will see there are two
types of current density in general
there are surface current density and
volume current density and from that
I'll move on to the continuity equation
which is one of the results of Maxwell's
equations so I'm going to first to find
the current per unit length and you
might understand it immediately but in a
moment I will be able to look at it in a
different way and perhaps that might be
a very used to you so just bear with me
for a moment we say that the current per
unit length is given by the placeholder
capital K now we also say that it is
equal to the surface charge density
multiplied by the velocity of the charge
now we know of course that if we
integrate the surface charge density da
we get the total charge so the first
formula for the the surface charge
density is equal to the serve excuse me
the surface current density is equal to
the surface cherish density multiplied
by the velocity of the charge now
another way of looking at this is saying
that capital K is the surface current
density and that it is it is the current
per unit width perpendicular to the flow
the important point to note here is that
it is perpendicular to the flow and I
will show you a diagram later on which
should illustrate this better so for the
moment I think the best way to think
about capital K is it that it is the
surface current density rather than
think of it thinking of it as the
current per unit width perpendicular to
the flow so if you think surface current
density you know that it comes from the
surface charge density multiplied by the
velocity of the charge
now let's try calculate the magnetic
force or the Lorentz force as a result
of this surface current density so we
know that the Lorentz force for for
magnetism is written on the top right of
your screen
and it's the surface integral of V cross
B DQ so the first thing I'm going to do
is sub in for DQ because we know the
total charge capital Q is the integral
of Sigma da so I'm going to plug in
Sigma da because we're integrating over
the surface so this is the magnetic
force due to surface current and we can
see pretty clearly that in fact we're
talking about
Sigma multiplied by V which of course is
the surface current density so we can
rewrite the magnetic force due to
surface current as the surface integral
of the surface current density crossed
which the magnetic field K cross P so in
this case if you can imagine that we
have a current flowing this way along
our cylinder the flow is going that way
so we take the the unit width
perpendicular to the flow so you can see
here I've defined the unit width
perpendicular to the flow now this is
one way of looking at it I prefer to
think of it as the surface charge
density multiplied by the velocity of
course they're all they're all that it's
the same thing really
now just to move on we know that current
itself is the flow of charge and we know
really what we're talking about is the
flow of electrons this gives us the unit
of coulombs per second as the unit of
the current now another way of thinking
of current is that it is the line charge
density lambda multiplied by the speed
of the charge so this should make
perfect sense to you having seen the
definition for the surface current
density the K is equal to Sigma V so
remember we used the surface charge
density here in order to define this so
we have done something similar here to
fighting the current as the line charge
density multiplied by the speed of the
charge or the velocity of the charge
note the unit's we have
line charge density of coulombs per
meter and speed of the charity m/s
giving us of course coulombs per second
so once again I'm going to try and
calculate the Lorentz force associated
with this particular at this particular
quantity so we want to calculate the
magnetic force of a line segment of
charge or basically a Wireless was
really so we integrate V cross B DQ but
we know that DQ is the line charge
density multiplied by the infinitesimal
length along the wire and that it's F
can be written in the following way so
we have the force due to a aligns
magnetic force due to a line segment
as the line integral of V cross B times
lambda da or we can write this in terms
of the current and in this way we can
write the following equation we have I
cross B integrated DL so this is the
magnetic force on a line segment now
it's important to note the following
let's say we have a wire I'm going to
I'm just going to draw a straight or a
reasonably straight wire in fact so this
is this is our wire but at any point we
want to take let's say this is the
current here the current is a vector so
as long as we take the small enough
segment R always going to get a straight
line but if you think about it as we
integrate we are also going to be
integrating along a straight line
segment so really what we're going to
see is that the excuse me the current is
always going to be perpendicular to the
immanent as well length segment so for
that reason we can exchange the current
and the infinitesimal length in the
integral and what we get is this final
equation here and this is the magnetic
force on a line segment so if the
current I outside of the line integral
of DL cross P so just compare this
moment with the same or dissimilar
integral we got a moment ago for the
magnetic force due to surface current
they're integrated the surface current
density crossed with the magnetic field
across the surface
here we integrate the line integral of
the current outside of D across be the
Internet has been line a line element
crossed with deep magnetic field so I'm
sure you can see what's going to come
next we've done two of the three
scenarios I'm going to look at and I'm
sure you can guess the next one we're
going to look at is volume charge
density so we give capital J the
placeholder for volume
excuse me volume current density yeah
Marcel TN capital J is the placeholder
for volume current density another way
of thinking that is the current per unit
area perpendicular to the floor now if
you just go to the bottom left of your
screen I have an illustration which
might help us here
so let's say we have cylindrical flow J
and it's flowing in this particular
direction here so the current is flowing
in that direction or the well of the
current density is flowing that way and
so as the current as a result but we
want to look at the density
perpendicular to the flow so for the
volume current density we look at the
the perpendicular area to the flow when
we looked at surface current density we
looked at the perpendicular length to
the flow here so I suppose in one you
can think of it as working one dimension
one dimension less but I think the
easier way to think about this is as the
volume current density so you could
probably suggest the following equation
even just by thinking about it that the
the volume current density is the volume
charge density multiplied by the
velocity of the charge now we know of
course from our study of electrostatics
is the total charge capital Q is the
integral of the volume charge density D
tau where D tau is there in fatass
amount of volume at a volume element so
putting that all together we find that
the volume current density is equal to
di da perpendicular so once again we're
going to try and calculate the magnetic
force on on a body due to the volume
current density so we start off the
Lorentz force on the top right of your
screen see if the volume integral of V
cross B D cube
Subban once again for DQ which is of
course Rudy Tao simply if then we can
sub in for the volume current density
capital J and we get that the magnetic
force magnetic force excuse me due to
volume current density is the volume
integral of the current density crossed
with the magnetic field integrated or
the volume of course they must remember
that magnetism is caused by moving
electric charge
not by stationary charges and we do not
have any magnetic monopoles or magnetic
charge themselves so I haven't once
spoken about the continuity equation but
that is coming next nonetheless though
the understanding or an understanding of
the volume current density the surface
current density in the line current
density is very important in our
understanding of the continuity equation
so just a little aside in regard to
units we know the current is the charge
multiplied by the velocity and that's
written on the top left of your screen
of course looking at all units we get
the units on charge or skimming the
units and current is coulombs per second
so let's just check the units if we look
at the current we have the line charge
density multiplied by the velocity of
the charge so that gives us coulombs per
meter multiplied by meters per second
giving us coulombs per second so this
equation for the line charge density
multiplied by the velocity is correct
so our line current formula is correct
what about our surface current well our
surface current is di e DL perpendicular
or another way of looking at it is the
is Sigma times the velocity where Sigma
is the surface charge density so the
product they are in terms of units is
going to be coulombs per meter squared
multiplied by meters per second giving
us coulombs per second per meter which
is exactly what we'd expect and finally
for volume current we know that the
volume current is equal to di da
perpendicular or another way of writing
it is Rho the volume charge density
multiplied by the
ozzie of the church once again we get
coulombs per meter cubed and meters per
meters per second giving us coulombs per
second per meter squared as anticipated
now we are ready to discuss the
continuity equation so a moment ago we
saw that the volume current density is
equal to di da perpendicular we can
rearrange this equation in the following
way saying that the infinitesimal
current element is equal to the volume
current density multiplied by the
infinitesimal area element perpendicular
to the flow of course then we can
integrate this to get the total current
and the total current is going to be
integral of G ETA perpendicular and the
usual trick here is to involve the dot
product because that picked up picks out
the perpendicular element so we get the
current is the surface at the surface
integral of G dot da now I'd like to
remind you of the divergence theorem
which are written in purple here so
basically in the divergence theorem we
go from a closed surface integral to a
volume integral so what I'm going to do
here is note that the moment we don't
have a closed surface integral for J dot
Da
so in fact I'm going to take a closed
surface integral and then invoke the
divergence theorem to be integrating
over the volume so we're going to have
the closed surface integral of G eta is
transformed to or as equivalent to
really as well as the volume integral of
the divergence of J now I hope you
realize that were actually looking at
here in terms of this surface integral
is the flux it's the flux of the volume
current density or the placeholder we
give for that is capital Phi but if
we're really talking about is the flow
of charge so the volume current density
comes from the volume charge density of
course if there is charge flowing out if
there's current flowing out of your body
it must happen or it must be due to a
flow of charge through the surface of
your body and it must come from the
inside so if there's charge or current
leaving your body it is coming as
the cost of charge our current inside
the body so that means we can look at
the flocks which is this the closed
surface integral off we'll say a dot da
that really should be J excuse me J dot
da it's going to be minus the volume
integral of the rate of change with
respect to time of volume charge density
integrated D town we could equate this
with the form that we have over here in
terms of the volume integral now a quick
inspection will convince you that the
integrand are equal and if the integrins
are to be equal that means that the
divergence of j is equal to minus del
Rho del T so that means that the
divergence of the volume current density
is minus the time rate of change of the
volume charge density we call this
equation the continuity equation and
although we haven't I haven't really
said that I have used nor invoked
Maxwell's equations you will see in
later videos when I discussed Maxwell's
equations that this an actual fact falls
straight out and you may already know
that we're we're invoking Maxwell's
equations even though I haven't directly
said so so the continuity equation is
really just an equation for the
conservation of charge so it's it should
make perfect sense to you that we have
the conservation of charge is one of the
fundamental laws so let's just move on
and have a I have a few comments first
of all I'd like to define what a steady
current is a steady current is one that
is never ending or it's never ending
flow of charge and what this means is
that there is no accumulation of charge
you never have an accumulation of charge
otherwise you wouldn't have studied
charges
now if skew steady currents if you have
steady currents we talked about magneto
statics
you might ask yourself why is that what
do you think about it
currents are the sources of the magnetic
force so steady currents will give us
the static magnetic forces so we talked
about magneto static steady currents
magneto statics and there at the Reedy
limit that we can reach of course of
never-ending flow and no accumulations
of charge however in many real
situations we can approximate what's
happening with this steady current and
therefore invoke the results of magneto
statics now it's interesting to note
that point charges do not constitute a
steady current because if they're here
one second they're gone the next and
that can't possibly be a steady current
there's one exception to this it's when
you're thinking of a let's say and very
much classical classical physics let's
think of a singular electron orbiting an
atom now the speed at which the electron
orbits the atom can often be on the
order of 10 to the 6 meters per second
which of course is a massive speed and
as a result it is equivalent or as close
to a steady current so although point
charges and steady currents electrons
orbiting an atom at that particular
speed they can often be approximated as
being a steady current now why have I
spent so much time discussing steady
currents well I'm about to tell you just
the same once more steady current to
something which is constant in magnitude
and has no buildup of charge by
definition if there is no built of
charge build-up of charge that means
Delta Rho or the change in the charge is
zero if the change in the charge is zero
that means the time rate of change in
the charge is going to be zero what this
implies is that the continuity equation
is itself zero
now remember what we're talking about in
this particular case we're talking about
magneto statics
so magneto statics in magneto statics we
get that the continuity equation is
equal to zero and of course this does
not hold from magneto dynamics or for
electrodynamics and this shouldn't
surprise you because for example we saw
with electrostatics that the curl of the
electrostatic field was equal to zero or
is equal to zero excuse me but the curl
of the vector dynamic field is nonzero
so there are certain situations where
you have a change in the equations or
the the simplification the equations
when you go from dynamic to the static
case so that's all I've got to say about
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