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184_notes:symmetry [2017/11/20 17:14] – created caballero | 184_notes:symmetry [2021/07/06 17:51] (current) – bartonmo | ||
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+ | Chapter 21 in Matter and Interactions (4th edition) | ||
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+ | [[184_notes: | ||
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===== Symmetry and Mathematical Tools ===== | ===== Symmetry and Mathematical Tools ===== | ||
- | One of the important aspects of electromagnetism is understanding the patterns of the electric and magnetic fields that the charges and current produce. These patterns can often suggest a symmetry -- a regular order to the pattern that helps us deal with the field. Symmetries are a deep part of physics. In this case, we are discussing geometric symmetries, which suggest that there' | + | One of the important aspects of electromagnetism is understanding the patterns of the electric and magnetic fields that the charges and current produce. These patterns can often suggest a symmetry -- a regular order to the pattern that helps us deal with the field. Symmetries are a deep part of physics. In this case, we are discussing geometric symmetries, which suggest that there' |
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+ | {{youtube> | ||
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+ | ===== Gauss' Law ===== | ||
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+ | [[184_notes: | ||
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+ | $$\Phi_E = \oint \vec{E} \bullet d\vec{A} = \dfrac{q_{enc}}{\varepsilon_0}$$ | ||
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+ | [[184_notes: | ||
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+ | $$\oint \vec{E} \bullet d\vec{A} = E \oint dA = \dfrac{q_{enc}}{\varepsilon_0}$$ | ||
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+ | [{{ 184_notes: | ||
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+ | The example we have seen a number of times is the point charge, $q$. If we encapsulate the point charge with an imaginary spherical surface of radius $r$, such that the point charge is at the center, we can easily find the electric field of the charge, | ||
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+ | $$\oint \vec{E} \cdot d\vec{A} = \dfrac{q_{enc}}{\varepsilon_0}$$ | ||
+ | $$E \oint dA = \dfrac{q}{\varepsilon_0}$$ | ||
+ | $$E 4\pi r^2 = \dfrac{q}{\varepsilon_0}$$ | ||
+ | $$E = \dfrac{q}{4\pi\varepsilon_0 r^2}$$ | ||
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+ | where we understand the direction to point radially outward as usual for a positive point charge. This was necessary to argue the simplification of Gauss' Law from the first to the second line. | ||
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+ | ===== Ampere' | ||
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+ | [[184_notes: | ||
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+ | $$\oint \vec{B} \bullet d\vec{l} = \mu_0 I_{enc}.$$ | ||
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+ | [[184_notes: | ||
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+ | $$\oint \vec{B} \bullet d\vec{l} = B \oint dl = \mu_0 I_{enc}.$$ | ||
- | ==== Gauss' Law ==== | + | [{{ 184_notes: |
+ | 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, | ||
+ | $$\oint \vec{B} \bullet d\vec{l} = \mu_0 I_{enc}$$ | ||
+ | $$B \oint dl \mu_0 I$$ | ||
+ | $$B 2 \pi r = \mu_0 I$$ | ||
+ | $$B = \dfrac{\mu_0 I}{2 \pi r}.$$ | ||
- | ==== Ampere' | + | where we understand the magnetic field to loop around the wire as given by the [[184_notes: |