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184_notes:examples:week5_flux_two_radii [2017/09/25 13:27] – [Solution] tallpaul | 184_notes:examples:week5_flux_two_radii [2021/06/04 00:43] – schram45 | ||
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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 | ||
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- | ===Approximations & Assumptions=== | ||
- | * 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. | ||
===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 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. | ||
+ | * Perfect spheres. | ||
+ | * Constant charge for the point charge (no charging/ | ||
+ | </ | ||
====Solution==== | ====Solution==== | ||
Before we dive into calculations, | Before we dive into calculations, | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
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: | 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: |