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| 184_notes:symmetry [2018/08/09 19:40] – [Gauss' Law] curdemma | 184_notes:symmetry [2020/08/24 19:26] – dmcpadden |
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| Chapter 21 in Matter and Interactions (4th edition) | Chapter 21 in Matter and Interactions (4th edition) |
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| [[184_notes:maxwells_eq|Next Page: Maxwell's Equations]] | /*[[184_notes:maxwells_eq|Next Page: Maxwell's Equations]] |
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| [[184_notes:conservation_theorems|Previous Page: Conservation Theorems]] | [[184_notes:conservation_theorems|Previous Page: Conservation Theorems]]*/ |
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| ===== Symmetry and Mathematical Tools ===== | ===== Symmetry and Mathematical Tools ===== |
| [[184_notes:motiv_amp_law|Ampere's Law]] helps us calculate the magnetic field when there is sufficient symmetry to use it. Like Gauss' Law, Ampere's Law, as a mathematical statement, is always true, but it's only useful in limited contexts (like long wires or solenoids). The total integral around any closed loop is always proportional to the total current enclosed by the loop, but that doesn't mean we can compute the magnetic field using Ampere's Law for every case, | [[184_notes:motiv_amp_law|Ampere's Law]] helps us calculate the magnetic field when there is sufficient symmetry to use it. Like Gauss' Law, Ampere's Law, as a mathematical statement, is always true, but it's only useful in limited contexts (like long wires or solenoids). The total integral around any closed loop is always proportional to the total current enclosed by the loop, but that doesn't mean we can compute the magnetic field using Ampere's Law for every case, |
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| $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}.$$ | $$\oint \vec{B} \bullet d\vec{l} = \mu_0 I_{enc}.$$ |
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| [[184_notes:loop|For a chosen Amperian loop]], if the magnetic field aligns with the direction of the loop and we can argue that the magnetic field is constant over the loop (or part of it and zero elsewhere), then we can simplify Gauss' Law sufficiently where it becomes useful, | [[184_notes:loop|For a chosen Amperian loop]], if the magnetic field aligns with the direction of the loop and we can argue that the magnetic field is constant over the loop (or part of it and zero elsewhere), then we can simplify Gauss' Law sufficiently where it becomes useful, |
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| $$\oint \vec{B} \cdot d\vec{l} = B \oint dl = \mu_0 I_{enc}.$$ | $$\oint \vec{B} \bullet d\vec{l} = B \oint dl = \mu_0 I_{enc}.$$ |
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| {{ 184_notes:week10_3.png?400}} | [{{ 184_notes:week10_3.png?400|Current through an amperian loop}}] |
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| The example that we have seen a number of times is the very long thin wire with current $I$. If we encircle the wire with a loop of radius $r$ with the wire centered inside the loop, we can easily find the magnetic field, | The example that we have seen a number of times is the very long thin wire with current $I$. If we encircle the wire with a loop of radius $r$ with the wire centered inside the loop, we can easily find the magnetic field, |
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| $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$ | $$\oint \vec{B} \bullet d\vec{l} = \mu_0 I_{enc}$$ |
| $$B \oint dl \mu_0 I$$ | $$B \oint dl \mu_0 I$$ |
| $$B 2 \pi r = \mu_0 I$$ | $$B 2 \pi r = \mu_0 I$$ |