# Homework Help: Flux Through a Cylinder

1. Dec 11, 2017

### RoyalFlush100

1. The problem statement, all variables and given/known data
In the attached image.

2. Relevant equations
Gradient(x, y, z) * <f, g, h> = <fx, gy, hz>

3. The attempt at a solution
Because the cylinder's not capped, I know that all the flux will be in the radial direction. So, I can find a normal vector by finding the gradient of the cylinder: n = <2x, 0, 2z>/(2sqrt(x^2+z^2)) = <x, 0, z>/sqrt(x^2+z^2)

Now, I want to put this in terms of t (the angle) and h (y):
r(t, y) = <acos(t), h, asin(t)>
Where: y: (-2, 2) and t: [0, 2pi)

Now we can rewrite the integrand:
<acos(t)/sqrt(a), 0, asin(t)/sqrt(a)> * <acos(t)/sqrt(a), 0, asin(t)/sqrt(a)> dS
=(a^2cos^2(t) + a^2sin^2(t))/a dS
=a dS

Now, the only thing I'm confused by (assuming everything else is right), is what to do with dS. I know it needs to be put in terms of dt and dh (where I already have the limits of integration), but I am unsure of how to perform this conversion.

#### Attached Files:

• ###### Flux.png
File size:
8.5 KB
Views:
140
2. Dec 11, 2017

Isn't the vector $\vec{E }$ of the field used in this flux equation simply a unit vector in the radially outward direction? If it is, (and I don't know that I should be giving you this much of the answer, but the problem is almost trivial), isn't the flux equal to the surface area?

3. Dec 11, 2017

### RoyalFlush100

I believe the answer should be a multiplied by the surface area, but I a unsure how to properly evaluate the integral, converting dS to be in terms of dt and dh.

4. Dec 11, 2017

You don't need to do any conversion. Surface area $S=\pi a^2 h$.

5. Dec 11, 2017

### RoyalFlush100

Okay. Just curious though, how would I perform that conversion in a less trivial case?

6. Dec 11, 2017

Surface integrals in the general case can take much effort to evaluate. The problem, I think, was designed to keep it simple.

7. Dec 11, 2017

### Orodruin

Staff Emeritus
In the general case, where you have a surface parametrised by the two variables $t$ and $s$, the directed surface element is given by
$$\newcommand{\dd}[2]{\frac{\partial #1}{\partial #2}} d\vec S = \vec n \, dS = \dd{\vec x}{t} \times \dd{\vec x}{s} dt\, ds,$$
where $\vec x$ is the position vector (which on the surface is parametrised by $t$ and $s$.