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184_notes:gauss_ex [2018/07/24 15:23] – curdemma | 184_notes:gauss_ex [2018/11/07 20:33] – dmcpadden | ||
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===== Putting Gauss' | ===== Putting Gauss' | ||
- | At this point, we have talked about how to find the electric flux through [[184_notes: | + | At this point, we have talked about how to find the electric flux through [[184_notes: |
{{youtube> | {{youtube> | ||
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$$\Phi_{tot}=\int \vec{E} \bullet \vec{dA}=\frac{Q_{enclosed}}{\epsilon_0}$$ | $$\Phi_{tot}=\int \vec{E} \bullet \vec{dA}=\frac{Q_{enclosed}}{\epsilon_0}$$ | ||
- | === Step 1 - Draw the electric field lines and determine a good Gaussian surface | + | === Step 1 - Draw the electric field lines === |
[{{ 184_notes: | [{{ 184_notes: | ||
- | // | + | // |
+ | |||
+ | === Step 2 - Determine a good Gaussian surface and find the electric flux through the Gaussian surface === | ||
+ | When we are picking our Gaussian surface, we want to pick a shape with sides that are either parallel to or perpendicular to the electric field vectors. In this case, a cylinder will work nicely. Remember that the choice of Gaussian surface is completely arbitrary, so we are picking a shape that will provide the simplest math. We will pick our cylinder to have a radius equal to $d=.05 m$, so that Point $P$ is on the edge of the cylinder. The height of the cylinder doesn' | ||
- | === Step 2 - Find the electric flux through the Gaussian surface === | ||
Now that we have a Gaussian surface, we can find the electric flux at the surface of the cylinder. The cylinder has three surface - the flat top, the flat bottom, and the curved side of the cylinder - so we need to account for the flux through all three surfaces to find the total electric flux. | Now that we have a Gaussian surface, we can find the electric flux at the surface of the cylinder. The cylinder has three surface - the flat top, the flat bottom, and the curved side of the cylinder - so we need to account for the flux through all three surfaces to find the total electric flux. | ||
$$\Phi_{tot}=\int \vec{E}_{top} \bullet \vec{dA}_{top}+ \int \vec{E}_{bottom} \bullet \vec{dA}_{bottom}+\int \vec{E}_{side} \bullet \vec{dA}_{side}$$ | $$\Phi_{tot}=\int \vec{E}_{top} \bullet \vec{dA}_{top}+ \int \vec{E}_{bottom} \bullet \vec{dA}_{bottom}+\int \vec{E}_{side} \bullet \vec{dA}_{side}$$ | ||
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$$\vec{E}_P =\frac{\lambda}{\epsilon_0 2\pi d} \hat{x}$$ | $$\vec{E}_P =\frac{\lambda}{\epsilon_0 2\pi d} \hat{x}$$ | ||
$$\vec{E}_P = 1439\: | $$\vec{E}_P = 1439\: | ||
+ | |||
+ | ==== Gauss' | ||
+ | Thus, to summarize, the steps for using Gauss' | ||
+ | - Figure out and draw the electric field around the charge distribution. | ||
+ | - Choose a Gaussian surface that a) goes through your observation point, b) has area vectors that are either parallel or perpendicular to the electric field vectors, and c) has a constant electric field along the Gaussian surface. This allows you simplify the electric flux integral. | ||
+ | - Find the amount of charge enclosed by the Gaussian surface (maybe using charge density if you need a fraction of the total charge). | ||
+ | - Solve for the electric field and determine the direction. | ||
+ | |||
==== Advantages and Disadvantages of Using Gauss' | ==== Advantages and Disadvantages of Using Gauss' |