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BOAS
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Homework Statement
Calculate the outward flux of [itex]F = 3z\mathbf{e_{\rho}} + cos(\phi)\mathbf{e_{\phi}} + 2z\mathbf{e_{z}}[/itex] through the base of a cylinder centered at the origin with radius [itex]\mathrm{R}[/itex] and height [itex]\mathrm{2H}[/itex].
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.
[itex]\Phi = \int_{s} \mathbf{F}. \mathbf{n} dA[/itex]
[itex]\mathbf{F}. \mathbf{n} = (3z\mathbf{e_{\rho}} + cos(\phi)\mathbf{e_{\phi}} + 2z\mathbf{e_{z}}).(\mathbf{e_{z}}) = 2z[/itex]
[itex]\Phi = \int_{s} 2z dA[/itex]
[itex]\Phi = \int_{s} 2z \rho d\rho d\phi[/itex]
[itex]\Phi = \int_{0}^{2\pi} \int_{0}^{R} 2z \rho d\rho d\phi[/itex]
[itex]\Phi = \int_{0}^{2\pi} z \rho^{2} d\phi = z \rho^{2} [\phi]^{2\pi}_{0}[/itex]
[itex]\Phi = 2 \pi z \rho^{2}[/itex]
[itex]\Phi = -2 \pi H R^{2}[/itex]
Thanks!
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