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Welcome back to our cause now it's time to solve another interesting example, another interesting problem,

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and this is a very famous problem it's cold calculating the magnetic fields of a charged wire.

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So as you see here, we have a wire that is trade and the current flows from the negative to the positive

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C direction in this blue wire here and already very early in school.

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We are told that the magnetic field of such a wire looks like this.

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So basically, it circulates around the wire.

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And if you look in the current direction, so basically from the top to the bottom, then the magnetic

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field will rotate along the clockwise direction.

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If you look from the top, it's of course turn counterclockwise.

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So that's maybe a bit of a similar example as the sphere that we have calculated previously.

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So the volume of the sphere, because the volume of the sphere and also the magnetic field of such a

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wire of both phenomena that we learned very early in school.

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However, these phenomena are quite difficult to drive and to explain in detail.

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But now, since we know the integrals, we can finally do it and here's how we do it.

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We need the Maxwell equations of Magneto aesthetics.

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So if you haven't heard about the Maxwell equations or Magneto statics, it's not a big problem.

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We only need these two equations here.

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So basically an integral equation and then also an equation that he uses to curl.

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So this is something that we learned previously in the previous section about the derivatives.

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So using the Maxwell equations of Magneto statics in their integral form, one can calculate the vector

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potential of a so-called current density distribution.

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So this word current density distribution is basically just a fancy word for all the currents in our

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system.

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So in our case, we just have a single current on the z axis.

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So everyone on the z axis, we have a current and this wire here, it has a radius and it has an area.

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If we cut through it and the density is, then just the current divided by the area of the cross section.

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And this will then be our j of our where this j of R will be non-zero only on the z axis, only in the

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wire, and it will be zero everywhere else because there are no currents.

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And then from this we can basically integrate over this term here.

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So this is something that one can derive.

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We will not go into the details here, but this is the equation you can derive and you have to integrate

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over the current density distribution divided by this term.

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Where does the absolute value of art, which is the position at which we look currently, for example,

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we can take a point in space this one here and we want to calculate the magnetic field at this particular

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point, and we have to integrate over these terms and we integrate over DX v Dash.

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And this means basically a triple integral to X, Y and Z.

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And this dash here, our dash basically means we integrate of all the points in space.

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And since the current density distribution is zero everywhere except for the wire, we only have to

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integrate over the wire in this case.

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So the R dashes are basically all the points of the wire.

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So when you think about it, and integral is just something like a sum.

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So you sum up this Ah, dash this our dash this, this this and so on and so on until you have the whole

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integral.

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And then we will, of course, plot this vector potential and look at it.

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However, the vector potential is not a real physical quantity that we can use or that we can measure.

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What we can measure is the magnetic field, and for this we have to calculate and the curl of the vector

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potential.

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And as I said, there are many more details here to consider.

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And if you find this topic interesting, you can have a look at my other course.

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It's called electrodynamics based on the Maxwell equations that we will really derive this equation,

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and we will also solve this particular problem here analytically.

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However, this course, we will solve it numerically, and this is how we do it.

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First of all, we must, of course, run the cell here.

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We must import Nampai and probably for plotting.

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And then we will consider our host trade wire, which is this blue wire here.

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And to solve the problem analytically, we would have to use some approximations.

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So now we solve it numerically, so we don't need these approximations, but I still want to be very

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close to the limits where we can.

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Ladies approximation, so then we can compare the analytical solution with our numerical solution.

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So first of all, we have to define some three factors.

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I will not care about the numbers here.

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I just set them to one.

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And then we have to define the wire, so we need the the current density.

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So as I said, current density will only be non-zero in the wire and zero everywhere else.

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So the current density will have a value j0 here in the wire and it will be constant in this j0 is,

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in our case, one.

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And if you would use a unit, it would be something like unpair per square meter.

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Then the wire has, of course, a radius, which is the limits where we can solve it analytically has

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to be very, very small, so it must actually be infinitely thin.

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So we take a small number here one millimeter and the length of the wire would be infinitely large in

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the limit where we could solve it analytically.

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So we need a very long wire, and I've just taken here one kilometer, which is actually very, very

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long.

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But OK, let's deal with such a long wire.

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So actually, the way that I will program here is that the wire has a length starting from minus L0

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going to plus L0.

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So actually, the length is two kilometers here.

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However, we will only consider the X y plane because if the wire would be infinitely large, then all

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other planes would behave equally in our case, since we have a finite length.

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It would, of course, depend a bit on the Z coordinate, but if we would say OK, we consider really

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an infinitely large wire, then we only have to consider a single plane and every plane would look the

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same.

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So we consider here a to Z equals zero plane.

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And then this one I told you already we have a very thin wire, so it has a radius are zero.

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But basically the current density will be non-zero.

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Only four X equals zero and Y equals zero, which is exactly the z axis.