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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 00:47] (current) – schram45 |
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| [[184_notes:eflux_curved|Return to Electric Flux through Curved Surfaces notes]] |
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=====Example: Flux through Two Spherical Shells===== | =====Example: Flux through Two Spherical Shells===== |
Suppose you have a point charge with value 1μC. What are the fluxes through two spherical shells centered at the point charge, one with radius 3 cm and the other with radius 6 cm? | Suppose you have a point charge with value 1μC. What are the fluxes through two spherical shells centered at the point charge, one with radius 3 cm and the other with radius 6 cm? |
* Φe for each sphere | * Φe for each sphere |
* d→A or →A, if necessary | * d→A or →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. | |
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===Representations=== | ===Representations=== |
→E=14πϵ0qr2ˆr
| →E=14πϵ0qr2ˆr
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* 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:5_flux_two_radii.png?300 |Point charge and two spherical shells}} | [{{ 184_notes:5_flux_two_radii.png?300 |Point charge and two spherical shells}}] |
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| <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. |
| </WRAP> |
====Solution==== | ====Solution==== |
Before we dive into calculations, let's consider how we can simplify the problem by thinking about the nature of the electric field due to a point charge and of the d→A vector for a spherical shell. The magnitude of the electric field will be constant along the surface of a given sphere, since the surface is a constant distance away from the point charge. Further, →E will always be parallel to d→A on these spherical shells, since both are directed along the radial direction from the point charge. See below for a visual. A more in-depth discussion of these symmetries can be found in the notes of [[184_notes:eflux_curved#Making_Use_of_Symmetry|using symmetry]] to simplify our flux calculation. | Before we dive into calculations, let's consider how we can simplify the problem by thinking about the nature of the electric field due to a point charge and of the d→A vector for a spherical shell. The magnitude of the electric field will be constant along the surface of a given sphere, since the surface is a constant distance away from the point charge. Further, →E will always be parallel to d→A on these spherical shells, since both are directed along the radial direction from the point charge. See below for a visual. A more in-depth discussion of these symmetries can be found in the notes of [[184_notes:eflux_curved#Making_Use_of_Symmetry|using symmetry]] to simplify our flux calculation. |
| |
{{ 184_notes:electricflux4.jpg?400 |Area-vectors and E-field-vectors point in same direction}} | [{{ 184_notes:electricflux4.jpg?400 |Area-vectors and E-field-vectors point in same direction}}] |
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Since the shell is a fixed distance from the point charge, the electric field has constant magnitude on the shell. Since →E is parallel to d→A and has constant magnitude (on the shell), the dot product simplifies substantially: →E∙d→A=EdA. We can now rewrite our flux representation: | Since the shell is a fixed distance from the point charge, the electric field has constant magnitude on the shell. Since →E is parallel to d→A and has constant magnitude (on the shell), the dot product simplifies substantially: →E∙d→A=EdA. We can now rewrite our flux representation: |