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184_notes:conservation_theorems [2017/11/20 21:17] – caballero | 184_notes:conservation_theorems [2017/12/01 00:13] – [Effects and Applications] dmcpadden | ||
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+ | Chapters 18 and 19 (and Chapters 2, 3, 6, 11, and 13) in Matter and Interactions (4th edition) | ||
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===== Conservation Theorems ===== | ===== Conservation Theorems ===== | ||
- | Conservation theorems are central to many aspects of physics: they often form the central reasoning principles for new observations, | + | Conservation theorems are central to many aspects of physics: they often form the central reasoning principles for new observations, |
$$\Delta \vec{p}_{sys} = \vec{F}_{ext} \Delta t$$ | $$\Delta \vec{p}_{sys} = \vec{F}_{ext} \Delta t$$ | ||
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$$\Delta \vec{L}_{sys} = \vec{\tau}_{ext} \Delta t.$$ | $$\Delta \vec{L}_{sys} = \vec{\tau}_{ext} \Delta t.$$ | ||
- | Electromagnetism is consistent with these fundamental principles (as you will see), but now that matter has charge, we bring a fourth fundamental principle to the party, | + | Electromagnetism is consistent with these fundamental principles (as you will see), but now that matter has charge, we bring a fourth fundamental principle to the party: [[184_notes: |
$$\Delta Q_{sys} = I_{ext} \Delta t.$$ | $$\Delta Q_{sys} = I_{ext} \Delta t.$$ | ||
- | These principles are referred to as " | + | These principles are referred to as " |
$$\Delta \vec{p}_{sys} = 0\, | $$\Delta \vec{p}_{sys} = 0\, | ||
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$$\Delta \vec{L}_{sys} = 0\, | $$\Delta \vec{L}_{sys} = 0\, | ||
$$\Delta Q_{sys} = 0\, | $$\Delta Q_{sys} = 0\, | ||
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+ | {{youtube> | ||
==== Linear and Angular Momentum Conservation in E&M ==== | ==== Linear and Angular Momentum Conservation in E&M ==== | ||
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+ | We have not talked much about linear and angular momentum conservation in an electromagnetic system because they extend beyond the scope of this course. This is because to truly understand the relationship between these and the electromagnetic field, we must develop an understanding that the electromagnetic field can have linear and angular momentum. That's right, the field itself has momentum that can push physical objects or twist them. This might seem very strange, but it is definitely the case that the electromagnetic field itself can have both. | ||
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+ | A common example of this comes from astrophysics. When a star is going through fusion, it has a lot of gas pushing outward from the core. In addition, light is carried outward. This is complicated process, but the gas and light run into material in front of them as they move towards the stellar surface. These pushes by the gas and light cause a pressure on the material in front of them; pushing them outward. However, the gas in front of the outward moving gas and light is gravitationally attracted to any matter behind it. This careful balance of the gravitational pressure, gas pressure, and radiation pressure (the momentum imparted by collisions of electromagnetic radiation with material) determines the size, temperature, | ||
==== Energy and Charge Conservation in E&M ==== | ==== Energy and Charge Conservation in E&M ==== | ||
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+ | Energy and charge conservation in electromagnetism is much easier to illustrate as both govern the movement of current in electronic circuits. In a typical circuit there are energy providers, [[184_notes: | ||
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+ | === Energy Conservation in a Circuit === | ||
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+ | A simple circuit consists of a single battery and a single resistor. There' | ||
+ | |||
+ | $$\Delta E_{sys} = W_{ext} + Q$$ | ||
+ | $$\Delta E_{battery} + \Delta E_{resistor} = q\Delta V_{battery} + q\Delta V_{resistor} = 0$$ | ||
+ | $$\Delta V_{battery} = -\Delta V_{resistor}$$ | ||
+ | |||
+ | where the minus sign indicates that the electric potential across the battery is negative as it is a user of energy. This calculation where we go around the loop adding up the energy per unit charge provided and used was [[184_notes: | ||
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+ | === Charge Conservation in a Circuit === | ||
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+ | Charge conservation in a circuit is a bit more subtle but explains how the current at any point in a simple circuit is the same. Consider a thick wire that narrows (our simple model for a resistor) and then expands again. In this situation, we found that the electrons in the thin part of the wire speed up to maintain a constant current. This occurs through the buildup of charge near the narrowing resistor causing a large gradient of surface charge near the resistor. | ||
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+ | We can apply charge conservation by choosing the resistor as our system. In this case, the amount of charge that builds up is zero as the system is in steady state, | ||
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+ | $$\Delta Q_{sys} = I_{ext} \Delta t$$ | ||
+ | $$\Delta Q_{sys} = 0 = I_{ext} \Delta t$$ | ||
+ | |||
+ | But now, let's care about the external currents, which are running into and out of the resistor. On the left side, the current goes in, but on the right side it goes out, so we will say that current is negative as it would act to reduce the charge in the system, | ||
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+ | $$0 = I_{ext} \Delta t = |I_{in}| \Delta t - |I_{out}| \Delta t$$ | ||
+ | $$|I_{in}| = |I_{out}|$$ | ||
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+ | Thus, the current going into the resistor, but be equal to the current coming out of it. We could choose any other part of the circuit like this and make the same argument, which means that charge conservation leads to an important result -- namely that the current into any branch is the same as that coming out. [[184_notes: | ||
==== Effects and Applications ==== | ==== Effects and Applications ==== | ||
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+ | Armed with these conservation theorems, namely energy and charge conservation, | ||
=== Resistors in a circuit === | === Resistors in a circuit === | ||
+ | {{ 184_notes: | ||
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+ | [[184_notes: | ||
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+ | $$R_{eq} = R_1 + R_2$$ | ||
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+ | {{184_notes: | ||
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+ | [[184_notes: | ||
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+ | $$\dfrac{1}{R_{eq}} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$$ | ||
=== Capacitors in a circuit === | === Capacitors in a circuit === | ||
+ | {{ 184_notes: | ||
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+ | [[184_notes: | ||
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+ | $$\dfrac{1}{C_{eq}} = \dfrac{1}{C_1} + \dfrac{1}{C_2}$$ | ||
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+ | {{184_notes: | ||
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+ | [[184_notes: | ||
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+ | $$C_{eq} = C_1 + C_2$$ |