Physics  1B  Electricity  &  Magne4sm  

Frank  Wuerthwein  (Prof)  Edward  Ronan  (TA)  

UCSD  

Outline  of  today  

•  Inductors  •  End  of  Chapter  23  •  Addi4onal  material  on  AC  currents  

Quiz  4  Content  •  Circular  and  rectangular  wires  and  coils  in  B-­‐fields  – Changing  B-­‐field  – Changing  Area  – Currents  given,  and  working  backwards  to  B-­‐field  or  area  changes  

•  What  is  inductance?  (Units,  defini4ons,  …)  •  Circuits  with  inductance  •  Sliding  wire  and  B-­‐field  (mo4onal  EMF  et  al.)  •  Faraday’s  law  

Energy  in  an  Inductor  •  We  say  that  a  capacitor  can  store  energy  in  its  electric  field  which  is  given  by:  

!

ECap = 12C "V( )2

"   We also say that an inductor can store energy in its magnetic field which is given by:

!

EL = 12 LI

2

"   If you attempt to change the current through the inductor, the energy stored in the inductor will perform work to oppose this change.

Concept  Ques4on  The  figures  to  the  right  show  three  circuits  with  iden4cal  baYeries,  inductors,  and  resistors.    Rank  the  circuits  according  to  the  current  through  the  baYery  a  long  4me  aZer  the  switch  is  closed  (greatest  first).  

A) 1, 2, 3.

B) 1, 3, 2.

C) 2, 1, 3.

D) 2, 3, 1.

E) 3, 2, 1.

2 1

3

End  of  Chapter  23  

AC  Circuits  •  An  AC  circuit  consists  of  a  combina4on  of  circuit  elements  and  an  AC  generator  or  source.  

•  An  AC  source  in  a  circuit  is  designated  by:  

"   In a simple circuit consisting of a resistor and an AC voltage source, the voltage drop across the resistor is equal to the voltage across the AC source.

AC  Circuits  "   The output of an AC generator is sinusoidal and varies

with time according to the following equation:

!

"v = "Vmax sin 2#ft( )"   where Δv is the

instantaneous voltage across the AC generator, ΔVmax is the maximum voltage of the AC generator, and f is the frequency at which the voltage changes (in 1/sec. = Hz)

AC  Circuits  "   We find that in a simple circuit consisting of a resistor

and an AC generator, that the voltage drop, ΔvR, across the resistor and current through the resistor, iR, are maximum values at the same time.

"   We say that the current and the voltage are “in phase” with each other.

"   If one were shifted to the left or right in any way, then they would be “out of phase.”

AC  Circuits  "   The direction of the current has no effect on the

behavior of the resistor.

"   To a resistor Imax in one direction is indistinguishable from Imax in the opposite direction.

"   The rate at which electrical energy is dissipated in the AC circuit is given by:

"   where i is the instantaneous current.

"   Note: Imax only occurs for a small amount of time, the instantaneous current is likely a different value.

!

P = i2R

AC  Circuits  "   Which average current value should you use for

calculating the energy dissipated?

"   The average value of current is zero.

"   But we actually see power in the circuit, so this can’t be right.

"   Since direction is not important, the current values should always contribute to power.

AC  Circuits  "   Squaring the current, will eliminate this directional

dependence.

"   So, we define a new value to help us. We call this the rms current: Irms = sqrt(time average of i2).

" rms stands for Root of the Mean of the Square.

!

Irms =Imax2

= 0.707( )Imax

AC  Circuits  "   The alternating voltage can also be discussed with

respect to rms values.

!

"Vrms ="Vmax2

= 0.707( )"Vmax

"   We can now define the average power dissipated by a resistor in an AC circuit carrying a current I as:

!

Pavg = Irms2 R

"   Ohm’s Law can either be expressed as:

!

"Vrms = IrmsR

!

"Vmax = ImaxR

AC  Circuits  with  Capacitor.  Consider  a  simple  circuit  containing  a  capacitor  and  an  AC  source.  

Ini4ally  the  current  charges  the  capacitor  plate  without  hinderance.  

"   This initial value of current will be a maximum.

"   As the charge on the plates increases, the voltage across the capacitor increases and the current flowing in the circuit decreases.

AC  Circuits  with  Cap.  "   The current then reverses direction.

"   Just before this reversal, the capacitor was still being charged; thus, its maximum charge and its maximum voltage drop occur as the current reverses direction.

"   The voltage across the plates decreases as the plates lose the charge they had accumulated.

"   The voltage across the capacitor, ΔVC, lags behind the current by 90o.

AC  Circuits  with  Cap.  "   What determines how much charge the capacitor gets (as well as

voltage drop)?

"   The capacitance, C, of the capacitor and the frequency, f, of the AC current.

"   Because of this we define a new variable which basically measures how much the capacitor resists current for a given AC source.

"   This variable is called capacitive reactance, XC, and is given by:

"   where f is measured in Hz and C is in F, this means that XC will be in Ω.

!

XC =1

2"fC=1#C

AC  Circuits  with  Cap.  "   Since capacitive reactance measures how a capacitor

resists current in an AC circuit, we can say that Ohm’s Law becomes:

"   What is the maximum current delivered to a circuit containing a 2.20μF capacitor when connected to an outlet having ΔVrms = 120V and f = 60.0Hz? !

"VC ,rms = IrmsXC

!

Imax = 2Irms

!

Imax = 2 "VC ,rmsXC

!

Imax = 2 120V( )2" 60Hz( ) 2.20 #10$6F( ) = 0.141A!

Imax = 2 "VC ,rms( ) 2#fC( )

Concept  Ques4on  A  capacitor  is  aYached  to  an  AC  voltage  source.    Which  change  will  result  in  a  doubling  of  the  rms  current  value?  

A) Halving the voltage and doubling the frequency.

B) Doubling the frequency.

C) Halving the frequency.

D) Doubling the voltage and halving the frequency.

AC  Circuits  with  Ind.  Consider  a  simple  circuit  containing  an  inductor  and  an  AC  source.  

Ini4ally  the  current  in  hindered  by  the  inductor’s  induc4on.  

"   Even though the initial value of current will be a maximum, the voltage drop will oppose it.

"   When the time rate of change of the current is maximum then the voltage drop across the inductor is a maximum.

AC  Circuits  with  Ind.  "   Also, when the time rate of change of the current is

zero then the voltage drop across the inductor is also zero.

"   Giving us the following:

"   As time moves on, we would first see a maximum voltage drop, then a maximum current.

"   The voltage across the inductor, ΔVL, leads the current by 90o.

For  Next  Time  (FNT)  "  Finish  reading  Chapter  23    

"  Finish  working  on  the  homework  for  Chapter  23