Solving Flux Integrals: A Step-by-Step Guide

[inner08]

Homework Statement

Well, I'm trying to take advance in my course and learn about how to calculate flux integrals. I have a few problems I can try out but I can't seem to understand the method they are using to solve the problems.

Homework Equations

I don't know..

All I have is int(surface) v x dA

The Attempt at a Solution

For example, I tried doing one of the problems. The question asks me to find the flux of the vectorial field through the surface.

F = yj through the square of sides 4 in the plane y=5. The square is centered on the y axis, its sides are parallel to the axis' and is oriented in the positive y direction.

Now I was trying to figure out some sort of method of how to solve problems like these.

This is what I did. I replaced y=5 in the equation F=yj and then multiplied it by the orientation which gave me F = 5. In this case, my dA was jdxdy.

From there, I had a double integral such as int(0,4) int(0,4) F x dA.

int(0,4) int(0,4) 5dxdz = 80

That does give me the correct answer but I want to know if the way I got to it is correct. If its not, are there any equations or other information that I should know about. None are given in the book :(.

Thanks!

 

[mighty2000]

Dear student,

It seems like you are on the right track with your approach to solving this problem. However, there are a few things that could be clarified.

Firstly, when you say "I replaced y=5 in the equation F=yj and then multiplied it by the orientation", it is not clear what you mean by "multiplied it by the orientation". It looks like you may have meant to write "multiplied it by the surface area", which would give you F = 20.

Secondly, I would suggest using the formula for flux through a surface, which is given by int(surface) F dot dA. In this case, since the surface is a square, the dot product between the vector field F and the normal vector dA will simplify to F_y*dA, where F_y is the y-component of F. This will give you F_y*Area, which is equivalent to what you have calculated.

Lastly, it is always a good idea to check your solution by using a different method, such as using the divergence theorem or a parametric representation of the surface. This will give you more confidence in your answer.

I hope this helps. Keep up the good work in your course!