Physics 2102 Lecture 17

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Physics 2102 Lecture 17

• DC circuits, RC circuits

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Where are we?
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Capacitors
Resistors
Series I  dQ/dt Same
Series Q Same
Parallel V Same
Parallel V Same
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Step II Apply Loop Rule
Around every loop add E if you cross a battery
from minus to plus, E if plus to minus, and iR
for each resistor. Then sum to Zero E1 E2 
 iR1  iR2   0.
Conservation of ENERGY!
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Step II Apply Junction Rule
At every junction sum the ingoing currents and
outgoing currents and set them equal. i1    i2
 i3
i2
i1
i3
Conservation of CHARGE!
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Step III Equations to Unknowns
Continue Steps IIII until you have as many
equations as unknowns!
Given E1 , E2  , i,R1 , R2  
E1 E2    i1R1   i2 R2   0 and i    i1
 i2 Solve for i2 , i3
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Example

• Which circuit has the largest equivalent
  resistance?
• Assuming that all resistors are the same, which
  one dissipates more power?
• Which resistor has the smallest potential
  difference across it?

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RC Circuits Charging a Capacitor
In these circuits, current will change for a
while, and then stay constant. We want to solve
for current as a function of time i(t). The
charge on the capacitor will also be a function
of time q(t). The voltage across the resistor
and the capacitor also change with time. To
charge the capacitor, close the switch on a.
E VR(t)VC(t) 0 E -i(t)R -q(t)/C 0 E -
(dq(t)/dt) R - q(t)/C 0
A differential equation for q(t)! The solution
is q(t) CE(1-e-t/RC) And then i(t) dq/dt
(E/R) e-t/RC
Time constantRC
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RC Circuits Discharging a Capacitor
Assume the switch has been closed on a for a long
time the capacitor will be charged with QCE.

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Then, close the switch on b charges find their
way across the circuit, establishing a current.
VRVC0 -i(t)Rq(t)/C0 gt (dq/dt)Rq(t)/C0
Solution q(t)q0e-t/RCCEe-t/RC i(t) dq/dt
(q0/RC) e-t/RC (E/R) e-t/RC
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Example

• The three circuits below are connected to the
  same ideal battery with emf E. All resistors have
  resistance R, and all capacitors have capacitance
  C.
• Which capacitor takes the longest in getting
  charged?
• Which capacitor ends up with the largest charge?
• Whats the final current delivered by each
  battery?
• What happens when we disconnect the battery?

Compile Rs into into Req. Then apply charging
formula with ReqC  ??
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Activity Time
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Example

• In the figure, E  1 kV, C 10 µF, R1 R2 R3
  1 MW. With C completely uncharged, switch S is
  suddenly closed (at t 0).
• Whats the current through each resistor at t0?
• Whats the current through each resistor after a
  long time?
• How long is a long time?

Compile R1, R2, and R3 into Req. Then apply
discharging formula with ReqC  ??
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Summary

• Technique to solve multiloop circuits
• Simplify compile circuits
• Apply loop rule
• Equations to unknowns
• RC circuits simple circuit with time-varying
  current time constant is t RC
• Charging a capacitor q(t) CE(1-e-t/RC)
• Discharging a capacitor q(t)CEe-t/RC