AP Physics “C” – Chapter 21 Notes – Yockers
Current and Direct Current Circuits
Current and Charge Movement
- current – the rate at which electric charges move through a given area
- current is the rate of charge movement
→current exists whenever there is a net movement of electric charge through a medium
Q
I
t
- I = current
- Q = amount of charge that passes through an area in a time interval
- t = time interval
→SI unit for current is the ampere, A
→1 A = 1 C/s
→instantaneous current – I the limit of the equation above as t goes to zero
Q dQ
I  lim

t 0 t
dt
- conventional current is defined in terms of positive charge movement
→moving charges that make up a current can be positive, negative, or a combination of
the two
→charge carriers – positive and negative charges in motion
→conventional current – current consisting of positive charges that would have the
same effect as the actual motion of the charge carriers
- structural model relating the macroscopic current to the motion of charged particles
Q  nAx q
Drift Velocity
- drift velocity – net velocity of a charge carrier moving in an electric field
Q  nAxd q  nAv d t q
- example – turning on a light
→flipping a switch establishes an electric field in wire
→electric field travels at nearly 3.00 x 108 m/s
→electric field causes charges to move
→charges do not travel at 3.00 x 108 m/s – can matter move at the speed of light?
- electrons do not travel in a straight line through conductor
→undergo repeated collisions with the vibrating metal atoms of the conductor
→result is a complicated zigzag pattern
→collisions increase vibrational energy of the atoms which causes the conductor's
temperature to increase
→energy gained by electrons as they are accelerated by the electric field is greater than
the average loss of energy due to the collisions
→thus, individual electrons move slowly along a conductor in a direction opposite of the
electric field
- drift speeds are relatively small
→electrons in copper wire with a current of 10 A move with a drift velocity of 2.46 x 10 -4
m/s (68 minutes to travel 1 meter!)
→electric field moves at approximately the speed of light
- alternative expression for current
Q
I
 nqvd A
t
- current density J in a conductor is defined as the current per unit area
I
J   nqv d
A
→J has units of A/m2
Sources and Types of Current
- difference in potential maintains current in a circuit
- batteries and generators supply energy to charge carriers – convert chemical energy to
Uelectric
- as charge carriers collide with the atoms of a device (light bulb or a heater) their electrical
potential energy is converted to kinetic energy
→electrical energy is used up, not charge
→batteries continue to supply electrical energy to charge carriers until chemical energy
is depleted
- current can be direct or alternating
→direct current (dc) – charges move in only one direction – light bulb emits light
continuously
→alternating current (ac) – motion of charges continuously changes in the forward and
reverse directions – light bulb "flickers" as current changes direction
- no net motion of charge
- frequency is 60 Hz (oscillates 60 times/second)
Behavior of Resistors
- drift speed is related to the electric field in a wire (linear relationship – drift speed is directly
proportional to the electric field)
- the potential difference across a conductor is proportional to the electric field (for a uniform
field in a conductor of uniform cross-section)
- when a potential difference is applied across the ends of a metallic conductor, the current in
the conductor is proportional to the applied voltage
→ I  V
→the equation above can be rewritten as V  IR , where R is the resistance of the
conductor
- some conductors allow charges to move through them more easily than others
- impedance of the motion of charge through a conductor is the conductor's resistance
→↑R→↓I
→↓R→↑I
- resistance – ratio of the potential difference across a conductor to the current it carries
V
R
I
→R = resistance
→SI unit for resistance is the ohm, 
→V of 1 V producing 1 A of current means the conductor has a resistance of 1 
- resistance is constant over a range of potential differences – Ohm's law
V
= constant
I
V  IR Ohm’s law (George Simon Ohm – 1787-1854)
→R is independent of V
→Ohm's law does not hold for all materials
- materials that have a constant resistance over a wide range of V are said to
be ohmic
- materials that do not function according to Ohm's law are said to be non-ohmic
→resistance depends on length, cross-sectional area, material, and temperature

R
A
- length increases → R increases
- cross-sectional area increases → R decreases
-  is the resistivity of the material (·m)
→depends on properties of the material and
→temperature
- the resistance of an ohmic conductor can also be expressed in terms of
conductivity    1   with units (·m)-1

R

A
Resistors in Series
- the current is the same in resistors in series because any charge that flows through R1
must also flow through R2
- the potential differences across the individual resistors is equal to the total potential
difference across the combination
Vtotal  V1  V2
V  IReq
and V  IR1  IR2  I R1  R2  so
V  IReq  I R1  R2   Req  R1  R2
- the equivalent resistance of a set of resistors connected in series is
Req  R1  R2  R3  ...
- the equivalent resistance of a series combination of resistors is the algebraic sum of the
individual resistances and is always greater than any individual resistance
- note that the r of an emf is in series with the Req of a circuit
Resistors in Parallel
- the potential differences across resistors in parallel are the same because each is
connected directly across the battery terminals
- the currents passing through parallel resistors are usually not the same (they can be)
- the charge entering a junction leading to parallel resistors must equal the “recombination” at
another junction of the charges leaving the resistors
I total  I 1  I 2
 1
V V
1  V
 

 V  
R1
R2
 R1 R2  Req
1
1
1


Req R1 R2
- the equivalent resistance of a set of resistors connected in parallel is
I
1
 1

1
1
Req   

 ...
 R1 R2 R3

- the inverse of the equivalent resistance of two or more resistors connected in parallel is the
algebraic sum of the inverses of the individual resistances and the equivalent
resistance is always less than the smallest resistance in the group
1
1
1
1



 ... or
Req R1 R2 R3
Electric Energy and Power
- as the charge of a system moves through a battery, whose potential difference is V, the
electrical potential energy of the system increases by the amount QV, and the
chemical potential energy in the battery decreases by the same amount (U = qV)
- as the charge moves through a resistor, it loses electrical potential energy during collisions
with atoms in the resistor
→the energy is transformed to internal energy corresponding to increased vibrational
motion of the atoms in the resistor
→energy transformation due to resistance is ignored (usually) in interconnecting wires
- the net result is that some of the chemical energy of the battery has been delivered to the
resistor which causes the resistors temperature to rise
- the charge (Q) loses energy (QV) as it passes through the resister, and if t is the time it
takes the charge to pass through the resistor, then the rate at which it loses electrical
potential energy is
dU d
dQ
 QV  
V  IV
dt
dt
dt
→I is the current in the resistor
→V is the potential difference across the resistor
- the rate at which energy is delivered to the resistor is
P  IV
→or, since V = IR for a resistor
2

V 
2
PI R
R
→this equation only applies to resistors and not to non-ohmic devices like light bulbs
and
diodes
- kilowatt-hour – the unit of energy used by electric companies to calculate consumption is
defined in terms of the unit of power and the amount of time it is supplied
→1 kWh = (103 W)(3600 s) = 3.60 x 106 J
Sources of emf
- a source of emf
→is any device that transforms nonelectrical energy into electrical energy
→the entity that maintains the constant voltage in a circuit
→any devices that increase the potential energy of a circuit system by maintaining a
potential difference between points in a circuit while charges move through it
- the emf  of a source is the work done per unit charge (SI unit is the volt, V)
- real batteries have internal resistance r, so the terminal potential difference is not equal to the
emf
- the terminal potential difference of the battery (V = Vb – Va) is
V    Ir
→ is equal to the terminal voltage (potential difference) when the current is zero (open
circuit voltage)
→r is the internal resistance of the battery and I is the circuit’s current
- a battery’s terminal voltage must also equal the potential difference across the external
resistance R (load resistance) or V = IR
  IR  Ir , solving for current gives
I

, and solving for total power gives
Rr
I  I 2 R  I 2 r
- if r<<R
→r can be neglected in the analysis of many circuits
→most of the power delivered by a battery is transferred to the load resistance
(resulting current will be small because the load resistance is large)
- if r>>R
→a significant fraction of the energy from the source of emf stays in the battery because
it is delivered to the internal resistance
Kirchhoff’s Rules and Simple DC Circuits
- Kirchhoff’s rules
→the sum of the currents entering any junction must equal the sum of the currents
leaving the junction (junction rule)
→the sum of the potential differences across all the elements around any closed circuit
loop must be zero (loop rule)
- the first rule is a statement of conservation of charge
- the second is a statement of conservation of energy
- problem-solving strategy
→draw the circuit diagram and assign labels to all known quantities and symbols to all
unknown quantities; assign directions to the currents in each part of the circuit;
although the assignment of current directions is arbitrary, one must adhere
rigorously to the directions you assign when Kirchhoff’s rules are applied
→apply the junction rule to all junctions in the circuit except one
→apply the loop rule to as many loops in the circuit as are needed to obtain, in
combination with the step above (junction rule), as many equations as unknowns;
must choose a direction to travel around the loop (see sign convention below)
→solve the equations simultaneously for the unknown quantities (if any resulting
currents have a negative value then the guessed “direction” was incorrect – the
current’s magnitude will be correct)
- sign convention (going from a to b)
- the current in a branch of a circuit containing a capacitor is zero under steady-state
conditions (the capacitor acts as an open circuit)
- to solve a particular circuit problem, you need as many independent equations as you
have unknowns (generally)
RC Circuits – Charging a Capacitor
- up to this point, all circuits discussed were considered to be steady-state circuits; circuits with
constant currents
- as a capacitor is charged by a battery through a resistor, the charge increases from zero to
some maximum value (which causes current to vary with time)
→charging continues until the capacitor is charged to its maximum equilibrium value
(Q = C, where  is the maximum potential difference across the capacitor)
→once the capacitor is fully charged, the current in the circuit is zero
-quantitative approach using Kirchhoff’s second rule
→potential difference across a capacitor
- “+”V going from the negatively charged plate to the positively charged plate
- “-“ V going from the positively charged plate to the negatively charged plate
→going clockwise around the circuit gives
q
   IR  0
C
- q and I are instantaneous values of charge and current as the capacitor is
charged
→the equation above gives the initial current in the circuit and the maximum charge on
the capacitor
- at t = 0 the charge on the capacitor is zero, and the initial current is a maximum
described by
I0 

the potential difference is entirely across the resistor
R
- later, when the capacitor is charged to its maximum value Q
→charges cease to flow
→the current in the circuit is zero
→the potential difference is entirely across the capacitor
Q  C (maximum charge)
- describing how the charge on a capacitor and current in a circuit vary with time
→remember that instantaneous current is I  dq dt , so
q
  IR 
C
C   IR   q
C  IRC  q
 IRC  q  C
 dq 
   RC  q  C
 dt 
dq
dt

q  C
RC
→the solution to the equation above is the time-dependent charge on a capacitor
dq
1

dt
q  C  RC
q
dq
1 t
0 q  C    RC 0 dt
t
 q  C 
ln 
ln y  x  y  e x

RC
  C 
q  C
 e t RC
 C
qt   C 1  e t RC  Q 1  e t RC
- e = 2.718 (not the fundamental charge)
- q = 0 at t = 0
- q approaches its maximum value Q as t approaches infinity
- the time constant ( = RC) represents the time it takes the charge on the
capacitor to increase from zero to 63.2% of its maximum value
→a capacitor charges very slowly in a circuit with a long time constant
→a capacitor charges very quickly in a circuit with a short time constant


 


→current can be found by differentiating the equation above with respect to time
x  e rt

dx
remember that I  dq dt ; I t   e t RC  I 0 e t RC
 re rt
R
dt
2
d x
 r 2 e rt
2
dt
→the potential difference across the capacitor at any time is
q
V 
C
RC Circuits – Discharging a Capacitor
- a circuit consisting of a capacitor with an initial charge Q, a resistor, and a switch will now be
considered
→the circuit above is the same as the one used for charging a capacitor except there is
no battery
→conditions before and after the switch is closed – loop rule
→potential difference across a capacitor
- “+”V going from the negatively charged plate to the positively charged plate
- “-“ V going from the positively charged plate to the negatively charged plate
→using the loop rule after the switch has been closed
q
  IR  0 , with q and I being instantaneous values
C
→the charge on the discharging capacitor is decreasing, so
- dq is negative, therefore
- the instantaneous current, I  dq dt , is also negative
- “negative” current flows opposite its original direction (the current while the
capacitor was charging)
q   dq 
 
R  0
C  dt 
q dq
 
R
C dt
dq
dt

q
RC
→this is a differential equation whose solution is the time-dependent charge on the
capacitor, so to determine charge as a function of time
t
dq
dt

Q q 0  RC
q dq
1 t
Q q   RC 0 dt
q
q
t
ln    
RC
Q
q (t )  Qe t RC
→capacitors lose charge exponentially with time (in one time constant, the capacitor
loses 63.2% of its initial charge; from Q to 0.368Q)
→differentiating the equation above with respect to time gives the current as a function
of time
dq
dQ t RC
(t ) 
e
dt
dt
I (t )   I 0 e t RC
→ V  et RC gives the decrease in potential difference, where  (Q/C), is the initial
potential difference across the fully charged capacitor