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184_notes:examples:week5_flux_two_radii [2017/09/18 12:35] – [Solution] tallpaul | 184_notes:examples:week5_flux_two_radii [2021/06/04 00:47] (current) – schram45 | ||
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+ | [[184_notes: | ||
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=====Example: | =====Example: | ||
Suppose you have a point charge with value $1 \mu\text{C}$. What are the fluxes through two spherical shells centered at the point charge, one with radius $3 \text{ cm}$ and the other with radius $6 \text{ cm}$? | Suppose you have a point charge with value $1 \mu\text{C}$. What are the fluxes through two spherical shells centered at the point charge, one with radius $3 \text{ cm}$ and the other with radius $6 \text{ cm}$? | ||
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* $\Phi_e$ for each sphere | * $\Phi_e$ for each sphere | ||
* $\text{d}\vec{A}$ or $\vec{A}$, if necessary | * $\text{d}\vec{A}$ or $\vec{A}$, if necessary | ||
- | |||
- | ===Approximations & Assumptions=== | ||
- | * There are no other charges that contribute appreciably to the flux calculation. | ||
- | * There is no background electric field. | ||
- | * The electric flux through the spherical shells are due only to the point charge. | ||
===Representations=== | ===Representations=== | ||
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$$\vec{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}$$ | $$\vec{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}$$ | ||
* We represent the situation with the following diagram. Note that the circles are indeed spherical shells, not rings as they appear. | * We represent the situation with the following diagram. Note that the circles are indeed spherical shells, not rings as they appear. | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
+ | |||
+ | <WRAP TIP> | ||
+ | ===Approximations & Assumptions=== | ||
+ | There are a few approximations and assumptions we should make in order to simplify our model. | ||
+ | * There are no other charges that contribute appreciably to the flux calculation. | ||
+ | * There is no background electric field. | ||
+ | * The electric fluxes through the spherical shells are due only to the point charge. | ||
+ | The first three assumptions ensure that there is nothing else contributing or affecting the flux through our spheres in the model. | ||
+ | * Perfect spheres: This will simplify our area vectors and allows us to use geometric equations for spheres in our calculations. | ||
+ | * Constant charge for the point charge: Ensures that the point charge is not charging or discharging with time. | ||
+ | </ | ||
====Solution==== | ====Solution==== | ||
- | Before we dive into calculations, | + | Before we dive into calculations, |
- | Since $\vec{E}$ is constant with respect to $\text{d}\vec{A}$ (in this case, it is sufficient that $\vec{E}$ is parallel to $\text{d}\vec{A}$ and has constant magnitude), we can rewrite our flux representation: | + | [{{ 184_notes: |
- | $$\Phi_e=\int\vec{E}\bullet | + | Since the shell is a fixed distance from the point charge, the electric field has constant magnitude on the shell. Since $\vec{E}$ is parallel to $\text{d}\vec{A}$ and has constant magnitude (on the shell), the dot product simplifies substantially: |
- | We can rewrite | + | $$\Phi_e=\int\vec{E}\bullet \text{d}\vec{A} = E\int\text{d}A$$ |
- | $$\left|\vec{E}\right| = \frac{1}{4\pi\epsilon_0}\frac{|q|}{r^2}\left|\hat{r}\right| = \frac{1}{4\pi\epsilon_0}\frac{|q|}{r^2}$$ | + | Note that $E$ is a scalar value representing |
- | We can plug in values for $q$ and $r$ for each spherical shell, using what we listed in the facts. For the smaller shell, we have $\left|\vec{E}\right|=1.0\cdot 10^7 \text{N/C}$. For the larger shell, we have $\left|\vec{E}\right|=2.5\cdot 10^6 \text{N/C}$. | + | $$E = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\left|\hat{r}\right| = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}$$ |
+ | We can plug in values for $q$ and $r$ for each spherical shell, using what we listed in the facts. For the smaller shell, we find $E=1.0\cdot 10^7 \text{ N/C}$. For the larger shell, we find $E=2.5\cdot 10^6 \text{ N/C}$. | ||
+ | To figure out the area integral, notice that the magnitude of the area-vector is just the area. This means that our integrand is the area occupied by $\text{d}A$. Since we are integrating this little piece over the entire shell, we end up with the area of the shell' | ||
+ | $$\int\text{d}A=A=4\pi r^2$$ | ||
+ | The last expression, $4\pi r^2$, is just the surface area of a sphere. We can plug in values for $r$ for each spherical shell, using what we listed in the facts. For the smaller shell, we have $A=1.13\cdot 10^{-2} \text{ m}^2$. For the larger shell, we have $A=4.52\cdot 10^{-2} \text{ m}^2$. | ||
- | In order to find electric flux, we must first find $\vec{A}$. Remember in the [[184_notes: | + | Now, we bring it together |
- | $$\vec{w}=3\text{ m }\hat{z}$$ | + | |
- | $$\vec{l}=5\text{ m }\cdot\cos 30^\circ (\hat{x})+5\text{ m }\sin 30^\circ\hat{y} = 2.5\sqrt{3}\text{ m } \hat{x} + 2.5\text{ m }\hat{y}$$ | + | |
- | Now, we can find the area vector: | + | |
\begin{align*} | \begin{align*} | ||
- | \vec{A} & | + | \Phi_{\text{small}} & |
- | | + | \Phi_{\text{large}} & |
- | &= 7.5\sqrt{3}\text{ m}^2 (-\hat{y}) + 7.5\text{ m}^2\hat{x} \\ | + | |
- | &= 7.5\text{ m}^2\hat{x} - 7.5\sqrt{3}\text{ | + | |
- | \end{align*} | + | |
- | Again, since the rectangle is not a closed surface, the choice for direction of $\vec{A}$ was arbitrary, and it would have been just fine to define it oppositely. Anyways, we can proceed to determine the electric flux: | + | |
- | \begin{align*} | + | |
- | \Phi_e &= \vec{E}\bullet\vec{A} \\ | + | |
- | | + | |
- | &= 60\text{ | + | |
\end{align*} | \end{align*} | ||
+ | |||
+ | We get the same answer for both shells! It turns out that the radius of the shell does not affect the electric flux for this example. You'll see later that electric flux through a closed surface depends //only// on the charge enclosed. The surface can be weirdly shaped, and we can always figure out the flux as long as we know the charge enclosed. |