How Do You Calculate Outward Flux Through the Base of a Cylinder?

[BOAS]

Homework Statement

Calculate the outward flux of F = 3z\mathbf{e_{\rho}} + cos(\phi)\mathbf{e_{\phi}} + 2z\mathbf{e_{z}} through the base of a cylinder centered at the origin with radius \mathrm{R} and height \mathrm{2H}.

Homework Equations

The Attempt at a Solution

I am unsure of how to tell if I have calculated the outward flux, and not entirely confident that it makes sense to have 'H' in my final answer. I'd really appreciate it if you could take a look.

\Phi = \int_{s} \mathbf{F}. \mathbf{n} dA

\mathbf{F}. \mathbf{n} = (3z\mathbf{e_{\rho}} + cos(\phi)\mathbf{e_{\phi}} + 2z\mathbf{e_{z}}).(\mathbf{e_{z}}) = 2z

\Phi = \int_{s} 2z dA

\Phi = \int_{s} 2z \rho d\rho d\phi

\Phi = \int_{0}^{2\pi} \int_{0}^{R} 2z \rho d\rho d\phi

\Phi = \int_{0}^{2\pi} z \rho^{2} d\phi = z \rho^{2} [\phi]^{2\pi}_{0}

\Phi = 2 \pi z \rho^{2}

\Phi = -2 \pi H R^{2}

Thanks!

 

[Dick]

Homework Statement

Calculate the outward flux of F = 3z\mathbf{e_{\rho}} + cos(\phi)\mathbf{e_{\phi}} + 2z\mathbf{e_{z}} through the base of a cylinder centered at the origin with radius \mathrm{R} and height \mathrm{2H}.

Homework Equations

The Attempt at a Solution

I am unsure of how to tell if I have calculated the outward flux, and not entirely confident that it makes sense to have 'H' in my final answer. I'd really appreciate it if you could take a look.

\Phi = \int_{s} \mathbf{F}. \mathbf{n} dA

\mathbf{F}. \mathbf{n} = (3z\mathbf{e_{\rho}} + cos(\phi)\mathbf{e_{\phi}} + 2z\mathbf{e_{z}}).(\mathbf{e_{z}}) = 2z

\Phi = \int_{s} 2z dA

\Phi = \int_{s} 2z \rho d\rho d\phi

\Phi = \int_{0}^{2\pi} \int_{0}^{R} 2z \rho d\rho d\phi

\Phi = \int_{0}^{2\pi} z \rho^{2} d\phi = z \rho^{2} [\phi]^{2\pi}_{0}

\Phi = 2 \pi z \rho^{2}

\Phi = -2 \pi H R^{2}

Thanks!

Seems ok. Except I would say the OUTWARD normal on the bottom of the cylinder is $-\mathbf{e_z}$. You could actually have done this calculation in your head, since you are integrating over a disk and the integrand is constant.

 

[BOAS]

Seems ok. Except I would say the OUTWARD normal on the bottom of the cylinder is $-\mathbf{e_z}$. You could actually have done this calculation in your head, since you are integrating over a disk and the integrand is constant.

Ah, of course! I should have picked up on that.

Thanks!