So let's first start with electric current.

Electric current is defined by

the motion of electrons or other charge with enough drift or flow.

And current density is the amount of charge passing per unit area and

per unit time through a surface element at right angles to the flow represented

by the vector j.

So simply put, current density can be thought of as

the electric current per unit area, but with a direction, okay?

And if we take a small area delta S at a given place in the material,

the amount of charge flowing across that area in a unit time is the j,

which is the current density, dot n which is the surface normal of

the surface of interest, times the area of that surface delta S,

where n is the unit vector normal to delta S.

So this this one will be the amount of charge.

So in other words,

this one will be delta q, okay?

So in the following slide, we'll discuss how this delta q is related to other

parameters we're going to discuss, especially the magnetic field, all right?

So in this slide, we will focus on current density j, which is a vector.

And the current density is related to the average flow of velocity of the charges.

Now, suppose that we have a distribution of charge whose average motion is

a drift with a velocity v as depicted on the right side here, so you'd have groups

of charge moving with uniform velocity v to the right side to Melodie, right?

And when it does happen you can see arbitrary chosen

area delta S sweeping a volume.

So, here I'm going to ask Melodie when that happens, how

much charge will be swept through the delta S inside dead volume.

>> Yeah, so if you think about this amorphous shape,

it all has the same density.

So when we sweep our surface through this volume that is

going to be moving at some velocity and some change in time.

And so we'll get a volume that's in the shape of a parallelogram and

then we can calculate the charges by this equation here.

So we know the density is remaining constant and

all that's changing is our surface moving at some velocity for some time.

And if you multiply these out, you can see that the velocity

times the time will give you the distance and then you multiply that by delta S

which will give you the volume so density times volume is equal to charge.

>> Exactly.

So as Melodie just explained to us, the parallelogram

has a volume of v delta t, multiplied by delta S.

And if you multiply volume by density, you would get the charge.

So that's how we get delta q.

Now, looking at this equation and

comparing with the equation we just discussed one slide before,

we now understand the current density j vector

is simply rho times v, which is velocity vector.

And as you know, rho is N, capital N, times q, so

Nqv will be the current density vector,

where capital N is the number of charge per unit volume.

Okay, so let's move on to current I.

We just covered the concept of j, which is a vector.

Now current is a scalar.

It is a scalar field.

And electric current is the total charge passing per unit time through

any surface S.

As you can see here from the equation, we are doing

surface integral of the current density to get our current.

So let's take a look at this picture.

You have a potato-shaped surface bounded by arbitrary loop and

if you think of current density going through this chip then you

can evaluate the current through the chip by doing this integration.

Now the current I out of a closed surface S

represents the rate at which charge leaves the volume V enclosed by S.

Because charge can neither be created nor annihilated,

which means conservation of charge is kept.

Then we can understand as you are picking more charge out of a out

of a jar, then you will have less and less charge left inside.

And the rate at which you are pulling out the charge will be the curve.

>> So one of the basic laws of physics is electric charge is indestructible and

is conserved.

In another words, in any closed surface, the outward current which is

the flux of the current is equal to the rate of the change.

The rate of decrease of your charge inside that volume.