184_notes:conservation_theorems

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184_notes:conservation_theorems [2021/07/06 17:36] – bartonmo184_notes:conservation_theorems [2021/07/06 17:36] (current) – bartonmo
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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 a complicated process, but the gas and light run into the material in front of them as they move towards the stellar surface. These pushes by the gas and light causes 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, and brightness of the star. Stellar formation and evolution is a vast research topic, but the point is that without that radiation pressure from the momentum carried by the electromagnetic radiation, the star could collapse under it's own gravity -- in fact, this is what happens in core-collapse supernovae! 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 a complicated process, but the gas and light run into the material in front of them as they move towards the stellar surface. These pushes by the gas and light causes 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, and brightness of the star. Stellar formation and evolution is a vast research topic, but the point is that without that radiation pressure from the momentum carried by the electromagnetic radiation, the star could collapse under it's own gravity -- in fact, this is what happens in core-collapse supernovae!
  
-==== Energy and Charge Conservation in E&M ====+===== Energy and Charge Conservation in E&M =====
  
 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:batteries|batteries]] and power supplies, and energy users or storers, [[184_notes:r_energy|resistors]] and [[184_notes:cap_charging|capacitors]]. If we consider the full circuit a closed system, there's is no energy transported out of it, so all the energy provided by things like batteries are used to drive currents in things like resistors. Conservation of charge occurs after steady state is reached, namely when the current is the same throughout the circuit. The charge built up to guide the current in the wires occurs quite quickly and once that is complete, there's no more charge buildup in the circuit unless the current begins to change with time (e.g., a new switch is closed, more batteries are added, a resistor is adjusted). 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:batteries|batteries]] and power supplies, and energy users or storers, [[184_notes:r_energy|resistors]] and [[184_notes:cap_charging|capacitors]]. If we consider the full circuit a closed system, there's is no energy transported out of it, so all the energy provided by things like batteries are used to drive currents in things like resistors. Conservation of charge occurs after steady state is reached, namely when the current is the same throughout the circuit. The charge built up to guide the current in the wires occurs quite quickly and once that is complete, there's no more charge buildup in the circuit unless the current begins to change with time (e.g., a new switch is closed, more batteries are added, a resistor is adjusted).
  
-=== Energy Conservation in a Circuit ===+===== Energy Conservation in a Circuit =====
  
 A simple circuit consists of a single battery and a single resistor. There's a little bit of energy used in the wires, but we can often neglect that amount compared to a resistor. The amount of energy provided to a single charge by the battery is just enough to make it around the circuit with most of that energy used in the resistor. We often model this using the amount of energy per unit charge ($\Delta V$, electric potential) provided by the battery as being used entirely by the resistor. This lets us avoid considering specific charges or combinations of charges.  A simple circuit consists of a single battery and a single resistor. There's a little bit of energy used in the wires, but we can often neglect that amount compared to a resistor. The amount of energy provided to a single charge by the battery is just enough to make it around the circuit with most of that energy used in the resistor. We often model this using the amount of energy per unit charge ($\Delta V$, electric potential) provided by the battery as being used entirely by the resistor. This lets us avoid considering specific charges or combinations of charges. 
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 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:r_energy#Energy_around_the_Circuit|called the loop rule]], which gave us a way to determine the current through a resistor (or other elements in a circuit).  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:r_energy#Energy_around_the_Circuit|called the loop rule]], which gave us a way to determine the current through a resistor (or other elements in a circuit). 
  
-=== Charge Conservation in a Circuit ===+===== Charge Conservation in a Circuit =====
  
 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. 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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 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:current|This was called the node or junction rule]]. 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:current|This was called the node or junction rule]].
  
-==== Effects and Applications ====+===== Effects and Applications =====
  
 Armed with these conservation theorems, namely energy and charge conservation, we can find some practical results for typical, passive circuit elements: resistors and capacitors. Armed with these conservation theorems, namely energy and charge conservation, we can find some practical results for typical, passive circuit elements: resistors and capacitors.
  
-=== Resistors in a circuit ===+===== Resistors in a circuit =====
 [{{  184_notes:week8_6.png?300|Resistors in series}}] [{{  184_notes:week8_6.png?300|Resistors in series}}]
  
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 $$\dfrac{1}{R_{eq}} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$$ $$\dfrac{1}{R_{eq}} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$$
  
-=== Capacitors in a circuit ===+===== Capacitors in a circuit =====
 [{{  184_notes:week8_12.png?300|capacitors in series}}] [{{  184_notes:week8_12.png?300|capacitors in series}}]
  
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