Story of $e^{x}$ and Flux

Electricity and Magnetism Level pending

A particle with some charge is position at $N (x, y, z)=(0,5,1)$ at that green dot as shown in figure. If the flux passing between the equations $e^{x}$ and $y=0$ for $x<0$ is $\phi_{1}$ and the flux passing between the equations $e^{x}$ and $y=0$ for $x>0$ is $\phi_{2}$ .Determine the ratio: $\frac{\phi_{2}}{\phi_{1}}$ .

The answer is 110.534.

1 solution

Steven Chase
Mar 15, 2020

My standard numerical integration approach doesn't work as well for this one, since the integration regions are infinite. So I outsourced the computations to Wolfram. I left out some of the scalar multipliers since they cancel out in the ratio (hence the "prime" superscripts).

$\phi'_1 = \int_0^{e^x} \int_{-\infty}^0 \frac{dx \, dy}{\Big(x^2 + (y-5)^2 + 1 \Big)^{3/2}} \approx 0.00818958 \\ \phi'_2 = \int_0^{e^x} \int_0^{\infty} \frac{dx \, dy}{\Big(x^2 + (y-5)^2 + 1 \Big)^{3/2}} \approx 0.90523$

Did it the same way. I like this problem, especially because the moment I saw it, I thought that the resulting integrals are likely to diverge. But this is not so, since the electric field magnitude also tends to zero as $x$ approaches infinity on either side of the number line, resulting in convergence. Interesting how the math works out here.

Karan Chatrath - 1 year, 2 months ago

I suppose the integrals can't diverge, because the total flux is a finite number (whatever would go through a sphere surrounding the charge). And of that, exactly half will go through the $xy$ plane. And then the fluxes under consideration are fractions of the flux going through the $xy$ plane.

Steven Chase - 1 year, 2 months ago

Oh, yes, that is a good argument. Had not thought of that.

Karan Chatrath - 1 year, 2 months ago

Story of e x e^{x} e x and Flux

The answer is 110.534.

1 solution

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Story of $e^{x}$ and Flux