Understanding the Cross Product and Its Role in Magnetism

[rockyshephear]

I just saw an instructor of EEE (YouTube) talk about Cross Product. He gave an example of when cross products are used. Here's what he said.

He drew a vertical arrow and said this is current. Then he drew a circular arrow around the vertical line and said it is B.

Then he drew a smaller vector, maybe unit, I'm not sure, on the vertical one and drew a vector from the base of the second vertical one to a point on the B arrow, stating that the cross product has something to do with these two vectors.

My question is since the current vector was not given a magnitude and neither was the B circle around the current arrow, how could one ever calculate the cross product.

AND since B technically goes on to infinity, at what diameter do you consider the end of the B vector to stop? If it's infinite, then I'd say the vector is infinite.

 

[Feldoh]

I'm not quite sure I'm understanding your description, but here's what I'm assuming:

The lecturer was trying to find the magnetic field as a function of position from the wire using the Biot and Savart Law: http://en.wikipedia.org/wiki/Biot–Savart_law in which a cross product is used. We use a cross product in this case because B is perdendicular to both the "position" vector and the "currect" vector.

The magnetic field is a function of position, so he was just finding the solution to the magnetic field for some arbitrary position relative to the wire.

However you are correct in some sense because in this case the B-field is continuous everywhere except a radial distance from the wire of 0 (r=0) so it does extend to infinity at which point B(+/- infinity) -> 0

 

[rockyshephear]

Could you not pick just an infinitely small point on the wire and such that any B in a plane perpendicular to that point could be a measurment of the magnetic field? Why must one pick a 'lenght' of wire?

Since the magnetic field is infinite, what properties of it make one different than another. The lines of flux will all be infinite. Can you determine a magnetic field in an infinitely thin plane that spreads out to infinity? Or do you need a certain thickness to determine the magnetic field?

I envision a magnetic field as a sheet that spreads to infinity, totally continuous but with every infinitely small point in that sheet having 'tendencies in certain directions' and 'strength in that direction' but still an infinitely thin sheet. Of course you can add sheet upon sheet until you reach infinity.

I'm just not getting the intuitiveness of what a field is.

And since the current flow thru the wire continuously, what difference does it make what section you pick?

Thanks for any help you can offer with these questions.

 

[Feldoh]

If you actually work out the Biot-Savart Law (BSL) for some finite wire, and then for some, presumed, infinite wire and you'll notice the magnetic fields are not the same. But using BSL you're summing up all the magnetic fields from infinitely many infinitely small points of a wire (since vectors are subject to super position)

Magnetic fields are vectors, so for different systems they will have different orientations and different strengths depending on the location.

I'm not sure what you're saying? A magnetic field is simply a vector field in euclidean space. Each point in space has a value and a direction for any given magnetic field.

 

[rockyshephear]

One of my questions was if the same current is a point x on the wire going thru it then the same magnetic field should exist at x plus 1 inch for sake of argument. There's nothing to produce anything but a homogenous set of vectors since the current is say 1 amp at point x and 1 amp at point x plus 1 inch. So how could there be a difference of the field at x or the field at x plus 1 inch. In fact thru the entire current loop the magnetic field should be the same.

 

[Phrak]

F = q (V x B) would be a better introductory example in electromagnetism.

 

[rockyshephear]

That doesn't mean much to me. What I know of that equation is as follows directly from the terms. My interpretation

Some nondescript force equals some charge multiplied times the cross product of the some voltage and some magnetic field. It really doesn't mean any more to me than that. How does it apply to the real world? Maybe the equation should be modified so it is more descriptive.

 

[Phrak]

If you just want an intuitive idea of the cross product without electromagnetism the best example is torque. There are plenty of web sites, and I see there are a couple utube lectures.

 

[rockyshephear]

I simply cannot get my question understood clearly. I must not be stating it properly. Allow me to try again, please. The guy in the video drew a vector from the current carrying wire, along the plane of the magnetic field. How did he know the magnitude of the vector. Could it be drawn a mm from the wire, an inch, a mile, a thousand miles? You can't very well draw a vector without knowing it's length. I guess I want to see someone set up a similar problem and explain how they know the magnitude of the vector along the wire and the magnitude of the vector in the plane of the magnetic field. And after doing so, do some kind of mathematical magic that solves some problem. What kind of question can be solved with these two vectors, if in fact one knew their respective magnitudes? I suppose I could reference the actual video and the approx time into the video that relates to my question.

 

[Feldoh]

The vector this guy drew from the wire to the magnetic field is arbitrary, it doesn't matter in the long run.

Basically the cross product just takes two vectors. Now, you can imaging these two vectors creating a plane in R^3 right? Well if you take the cross product of the two vectors it gives you a vector that is perpendicular to the first two vectors that you crossed. Or normal to the plane, if you will.

 

[rockyshephear]

Sure. I get that any two vectors with real magnitudes and directions, form a paralellliped, [|vector a| times |vector b| times cos theta] which is the magnitude of a new vector parallel to the plane of the two vectors. What does this have to do with current going thru a wire causing a magnetic field? If you can answer this, maybe that will answer my question. Thx

 

[Phrak]

You could have found this with a simple search

$$\bar{B} = \mu_{0} \bar{I} \times \bar{R} / 2 \pi R^2$$

 

[rockyshephear]

Can you provide a drawing that shows what these terms mean? While I see two vectors that could be cross producted, what determines the value of R? I obviously is the current but I would have no idea what to put in for R.

 

[dx]

Can you provide a drawing that shows what these terms mean? While I see two vectors that could be cross producted, what determines the value of R? I obviously is the current but I would have no idea what to put in for R.

R is just the distance from the wire to the point at which you are calculating the magnetic field

 

[rockyshephear]

Let's see. So R is a vector that points perpendicular to the line of current whose magnitude ends at the distance of the point you are determining the magnetic field for.

There should be another term for the magnetic field at a point because a point does not represent a field. A field is a large collection of points. Magnetic field seems to imply your describing the entire field, not a point in the field.

So now we have R which is a vector but I still don't get why the current is a vector. It certainly has a direction if the wire is straight but it has no magnitude. In my mind the current cannot be a vector because you are determiining the perpendicular R vector from a specific POINT on the wire, not from a length of wire. And since the current is constant, it should make no difference if you consider a point on the wire or a mile of the wire. The magnetism is measure from some point outward perpendicular to the point to the R point of measurement. It doesn't matter if the I point is at the beginning of the inch of wire or at the end of an inch of wire. It's just a point. So why is length of the current carrying wire important. Please don't just spit out an equation to answer the question. Thanks.

 

[Mute]

A magnetic field is a vector field: at each point in space there exists a vector describing the magnitude and direction of the field at that point.

Current can be given a vector: you just multiply the magnitude by a unit vector in the direction of the current flow. Current certainly has a magnitude. You can specify the value of the current at any point in the wire, just as you can specify at projectile's velocity at any point on its trajectory. Typically, though, the value of the current is constant, so we don't need to worry about it changing in time and so a single vector describes the current anywhere along the wire.

Then, you consider a position vector from the wire to any point of interest, and take the cross product. The result is a vector proportional to the magnetic field at that point. If you did this computation at every point, you would have computed the entire vector field describing the magnetic field. If the current depended on time then so would the magnetic field.

 

[Phrak]

Let's see. So R is a vector that points perpendicular to the line of current whose magnitude ends at the distance of the point you are determining the magnetic field for.

You're right. I was wrong. It's junk. At best, it's a very conditional cross-product.