184_notes:conservation_theorems

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision		 Previous revision
Next revisionBoth sides next revision
184_notes:conservation_theorems [2017/11/30 01:29] – dmcpadden184_notes:conservation_theorems [2020/08/24 19:26] – dmcpadden
Line 1: Line 1:
 Chapters 18 and 19 (and Chapters 2, 3, 6, 11, and 13) in Matter and Interactions (4th edition) Chapters 18 and 19 (and Chapters 2, 3, 6, 11, and 13) in Matter and Interactions (4th edition)
 +
 +/*[[184_notes:symmetry|Next Page: Symmetry and Mathematical Tools]]
 +
 +[[184_notes:magnetic_field|Previous Page: The Magnetic Field]]*/
  
 ===== Conservation Theorems ===== ===== Conservation Theorems =====
Line 19: Line 23:
 $$\Delta \vec{L}_{sys} = 0\,\mathrm{when}\, \vec{\tau}_{ext} = 0$$ $$\Delta \vec{L}_{sys} = 0\,\mathrm{when}\, \vec{\tau}_{ext} = 0$$
 $$\Delta Q_{sys} = 0\,\mathrm{when}\, I_{ext} = 0 $$ $$\Delta Q_{sys} = 0\,\mathrm{when}\, I_{ext} = 0 $$
 +
 +{{youtube>N5s0mi7BV6g?large}} 
  
 ==== Linear and Angular Momentum Conservation in E&M ==== ==== Linear and Angular Momentum Conservation in E&M ====
Line 38: Line 44:
 $$\Delta V_{battery} = -\Delta V_{resistor}$$ $$\Delta V_{battery} = -\Delta V_{resistor}$$
  
-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 called the loop rule and it 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 in any branch of a 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.
  
 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, 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,
Line 54: Line 60:
 $$|I_{in}| = |I_{out}|$$ $$|I_{in}| = |I_{out}|$$
  
-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. 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 ====
  
Line 60: Line 67:
  
 === Resistors in a circuit === === Resistors in a circuit ===
 +[{{  184_notes:week8_6.png?300|Resistors in series}}]
  
-Two resistors in series (end-to-end) must have the same current running through them, but they can use different amounts of energy per unit charge to drive that current depending on their individual resistances. This leads to their combined, effective result on the current in a circuit increasing the resistance of the circuit,+[[184_notes:r_series|Two resistors in series]] (end-to-end) must have the same current running through them, but they can use different amounts of electric potential to drive that current depending on their individual resistances. This leads to their combined, effective result on the current in a circuit increasing the resistance of the circuit,
  
 $$R_{eq} = R_1 + R_2$$ $$R_{eq} = R_1 + R_2$$
  
-Two resistors in parallel (connected off the same branch) must use the same energy per unit charge, but they can drive different currents as long as the sum of those currents is equal to the total before the branch splits. This leads to their combine effective result as reducing the overall resistance of the circuit,+[{{184_notes:week8_9.png?400|Resistors in parallel  }}] 
 + 
 +[[184_notes:r_parallel|Two resistors in parallel]] (connected off the same branch) must use the same electric potential (by the loop rule), but they can drive different currents as long as the sum of those currents is equal to the total before the branch splits. This leads to their combine effective result as reducing the overall resistance of the circuit,
  
 $$\dfrac{1}{R_{eq}} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$$ $$\dfrac{1}{R_{eq}} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$$
  
 === Capacitors in a circuit === === Capacitors in a circuit ===
 +[{{  184_notes:week8_12.png?300|capacitors in series}}]
  
-Two fully charged capacitors in series (end-to-end) must have the same total charge, otherwise a current would be driven until such time that they did. This means that each of them can store a different amount of energy per unit charge, which depends on their individual capacitance. The result is reducing the overall capacitance of the circuit,+[[184_notes:c_series|Two capacitors in series]] (end-to-end) must have the same amount of stored charge, otherwise a current would be driven until such time that they did. This means that each of them can store a different amount of energy per unit charge, which depends on their individual capacitance. The result is reducing the overall capacitance of the circuit,
  
 $$\dfrac{1}{C_{eq}} = \dfrac{1}{C_1} + \dfrac{1}{C_2}$$ $$\dfrac{1}{C_{eq}} = \dfrac{1}{C_1} + \dfrac{1}{C_2}$$
  
-Two fully charged capacitors in parallel (connected off the same branch) must use the same energy per unit charge, but they can store different amounts of charge depending on their individual capacitances. The result is increasing the overall capacitance of the circuit,+[{{184_notes:week8_15.png?400|Capacitors in parallel  }}] 
 + 
 +[[184_notes:c_parallel|Two capacitors in parallel]] (connected off the same branch) must use the same energy per unit charge (because of the loop rule), but they can store different amounts of charge depending on their individual capacitances. The result is increasing the overall capacitance of the circuit,
  
 $$C_{eq} = C_1 + C_2$$ $$C_{eq} = C_1 + C_2$$
  • 184_notes/conservation_theorems.txt
  • Last modified: 2021/07/06 17:36
  • by bartonmo