# Feat. Oliver, Evan, Michelle, and John.  Material property ρ  Describes the microscopic structure of a conductor ▪ The stuff that an electron has to.

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feat. Oliver, Evan, Michelle, and John

 Material property ρ  Describes the microscopic structure of a conductor ▪ The stuff that an electron has to move through  Total resistance depends on resistance  What makes it harder for an electron to move? ▪ Longer wire ▪ Less cross-sectional area R=ρL/A

 Deals with potential difference, resistance, and current  Current (speed which charges can move) is directly related to how much they want to move (voltage) and inversely related to how much stuff is in their way (resistance) I=V/R V=IR

 Kirchhoff’s Voltage Law (KVL) (Loops)  Around a closed loop, the sum of all voltage changes must equal zero  Kirchhoff’s Current Rule (KCL) (Nodes)  The sum of currents flowing into a node equals the currents flowing out of a node

 In series, current through each resistor is equal (no node splits), but voltage is split up, so… ΔV TOT = ΔV 1 + ΔV 2 IR TOT =IR 1 +IR 2 R TOT =R 1 +R 2

 In parallel, voltage drop is the same (closed loops all over), but current is split up, so… I TOT =I 1 +I 2 V/ R T =V/ R 1 + V/ R 2 1/R T =1/R 1 +1/R 2

 P= ΔW/ΔtΔW=(Δq)VI=Δq/t P=IV  Using V=IR, we can substitute to get:  P=IV=(V/R)(V)=V 2 /R  P=IV=(I)(IR)=I 2 R

Math is on the handout.

 Remember C=Q/V  Capacitors in Parallel follow KVL  C T =C 1 +C 2  Capacitors in Series follow KCL  1/C T =1/C 1 +1/C 2

 Remember I=dQ/dt; C=Q/ ΔV  Apply KVL:  Over time, the amount of current in the circuit decreases

 τ=RC  Time constant for charging depends on resistance in the circuit and how much the capacitor can hold  At 3τ, the device is 95% charged ▪ Steady state conditions  Meanwhile while charging…  I asymptotically approaches 0 as I =  Q stored asymptotically increases to CV battery  V cap approaches V battery

 Rearranging gives you  Constants depend on charge in circuit at t=0 and t=infinity  But what really matters:  At first, there is no charge in a capacitor. Over a long period of time, the charge equals the product of capacitance and battery voltage

 From previous capacitor info  U=1/2(QV)=1/2(C V 2 )=1/2( Q 2 /C)

 Current in the circuit is defined as: V here is the V of the cap, not necessarily the V of supply  Charge in the Capacitor is defined as: Q0 is the charge on the capacitor when the switch is flipped

 A capacitor of capacitance C is discharging through a resistor of resistance R. Leave answers in terms of τ.  When will the charge on the capacitor be half of its initial value?  When will the energy stored in the capacitor be half of its initial value?

 Half charge?  Using discharging formula  Adjust for half  Take natural log  Rearrange for t

 Half energy?  Using energy formula  Adjust for discharging  Adjust for half  Take logs  Rearrange for t U= 1/2( Q 2 /C)

 So in case you haven’t been paying attention, what is an RL circuit?:  As the name would imply it is a circuit with a resistor and inductor. ▪ What is a resistor?: It resists things. In the case of circuits it is resisting the flow of electrons. ▪ What is an inductor?: A circuit component that initially tries to oppose changes in the current running through it. After a while it stops fighting the change and begins acting like a wire.

 So in an RL circuit that you just turn on the current in the resistor does not rise quickly to a value of ε/R, but it doesn’t. Why? ▪ The inductor in the circuit creates a self induced emf to oppose the rise of the current, which in turn means that the emf of the inductor is opposite in polarity to that of the battery. ▪ As long as the inductor is opposing the emf of the battery the resistor will have a value smaller than ε/R. As time goes on the discrepancy between what the current at the resistor should be and what it actually is will get smaller.

 But Evan, what kind of formulas could we use in solving RL circuits? Well, I’m glad you asked: Loop rule for RL circuits.

 But what if I want more forumals?:

 A circuit consisting solely of an inductor and a capacitor  In the circuit, stored energy oscillates at a specific resonant frequency

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