I’m taking an introductory course of physics in college (my major has nothing to do with maths nor physics; this is just an elective) and it came out while going over magnetic flux. Any google search I make ends up with some complicated maths explaining what it is which I don’t understand. What does it do, what is it, and why does my square have a vector coming out from the middle of it?
In: Mathematics
It basically tells you in which direction the area DOESN‘T expand. As a plane area is 2-dimensional you either need two directions to tell where the area is in 3-dimensional space. Or you tell one direction it doesn’t go, each direction that’s perpendicular to that one is inside the area.
Short ELI5: it‘s useful for knowing where the area is and isn’t.
This appears to be a really obscure thing. I’ve never heard of it in any physics class, and the internet isn’t exactly ripe with good answers either. Merriam-Webster has this definition:
> the vector of a plane surface whose magnitude is the area of the figure and whose direction is that of a perpendicular to the plane of the figure
What this means is: It is a vector (an arrow or just a line) sticking out of a completely flat surface at a 90° angle (often called “normal” vector), and its length tells you the area of the surface.
Edit: I have absolutely no idea for what this could be useful.
The direction of it is just perpendicular because that’s the best way to indicate the dimension that the area is in.
In the case of physics also allows for dot products with vector fields, which is a way to multiply vectors to get a component that points in the same direction as one of the vectors. Basically, a field can be visualized as a bunch of lines going into/out of a point (a charge or a magnet). Then, the amount of those lines poking through an area is a quantity called the flux. But, you only want the part perpendicular to the area to count towards the flux (harder to explain, but it’s basically that if the field is at a non-right angle to the area, there’s less force pulling through the area). So, you can take the field and dot product it with the area vector, which we defined as pointing perpendicular to the area, to get the amount of field passing through that area.
This quantity flux has a lot of uses in physics, the most important being Gauss’s law, which says that the amount of flux through a surface area is proportional to the amount of charge enclosed by the surface. Physicists/physics students will take the flux through something like an imaginary sphere surrounding a point charge, and use it to determine how much charge that point has
addendum to previous answers:
the length of the vector in scalar (aka number) units is equal to the area of the plane in units of area. this vector is easily calculated by computing the cross-product of two vectors along the sides of the plane spanning up the whole plane. it doesn’t matter whether these side vectors are perpendicular or have the same length, the resulting cross product will always be a vector with said properties.
the gist is, you can describe a plane (orientation and area) with just one vector, so the information gets condensed down to the minimum amount of numbers
The magnetic flux through an area is dictated by the size of the area, how strong the magnetic field is, and whether or not the field is passing through the area. So to calculate the flux we need to know not only how big the area is, but how it’s oriented relative to the magnetic field. The easiest way to do this mathematically is to use a vector because a vector has both of those things: size and direction.
So what we do is to say the direction of the area vector is normal (perpendicular) to the surface, and its magnitude is the area. If we do that then the equation for flux ends up pretty simple: flux = ABcos(theta), where A is the area vector, B is the field, and theta is the angle between the two. The math here should make some intuitive sense when you think about it. If the normal of the area plane is lined up with the field direction, you have cos(theta) = 1, which gives you the maximum value. If the field is perpendicular to that plane you get cos(theta) = 0, which means no flux at all, because the field isn’t going through the plane. [Here’s a picture of those two circumstances](https://qph.fs.quoracdn.net/main-qimg-c943672572b7f118c83c6135e47daa40).
> why does my square have a vector coming out from the middle of it?
That’s the area vector. It’s normal (perpendicular) to the area, because a normal vector is the only single vector that tells you how the plane is oriented.
Suppose I have a solar panel, and I want to find the amount of sunlight that shines on it. Well obviously that depends on the brightness of the sunlight and the area of the panel, but it also depends on the *angle* of the panel. If it’s face-on to the sun, that’s good, but if it’s edge-on to the sun, the light will go above and below the panel rather than striking it, so I won’t get any solar power. If my panel’s at an angle to the sun, then only the component of the sun’s rays pointing *into* the panel matters.
I can calculate this by drawing an imaginary arrow perpendicular to the face of the panel, whose magnitude equals the area of the panel — this is the solar panel’s area vector. For maximum solar power, we want the sunlight to be parallel to this area vector. The solar energy input is the dot product of the sunlight vector with the area vector, because the dot product is all about two vectors being parallel or not.
Victor? What do you say…
The length of the vector in scalar is equal to the area of the square. And it’s exactly perpendicular to the area itself. The most useful thing about it, is that the equation of the plane on which the area rests is actually the direction of the vector itself equal to 0. It makes finding the plane equation super easy.
Latest Answers