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Direct Chapter Current Circuits
1. 2. 3. 4. 5. R connections in series and in parallel Define DC (direct current), AC (alternating current) Model of a battery Circuits with 2+ batteries Kirchhoff's Rules RC circuit
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Confucius says, reviewing helps one learn new knowledge.
Concepts:
Charge: positive, negative, conserve, induction. potential Electric: field flux Electrostatic equilibrium: no moving charge. Current: moving charge Capacitance: potential over charge Resistance: potential over current Resistance and Resistivity (conductivity) and temperature:
Laws:
Coulomb's: force and charge.
Gauss's: Ohm's:
electric flux and charge. electric potential and current.
Circuits and components:
Symbols: Power: Capacitor: Resistor: wire, battery, C, Switch... current time potential difference: in parallel in series discuss today
Resistor connections
In series. Condition:
Req V I V1 I
V1
V2
I
V2
I1 I2 V
V1 I1
V1
V2 I2
V2
R1 R2
In parallel. Condition:
1 Req I V I1 I2 V
I
I1 I2 V
I1 V1
V1
V1
I2 V2
V2
1 R1 1 R2
V2
Resistor connections
In series, I I1 I2 : voltage sharing power sharing
V1 V2 P1 P2 R1 R2 R1 R2
V1
V2
In parallel, V V1 current sharing
I1R1 P1R1
V2 :
R2 R1 R2 R1
power sharing
I1 I2R2 , or I2
P1 P2R2 , or P2
Resistors connections, summary
In series
Req R1 R2 R3 ...
V1 V2 P1 P2
R1 R2 R1 R2
In parallel
1 Req 1 R1 1 R2 1 R3 ...
I1R1 I2R2 PR1 P2R2 1
Resistors in Series Example
Use the active figure to vary the battery voltage and the resistor values Observe the effect on the currents and voltages of the individual resistors
PLAY ACTIVE FIGURE
Resistors in Parallel Example
Use the active figure to vary the battery voltage and the resistor values Observe the effect on the currents and voltages of the individual resistors
PLAY ACTIVE FIGURE
Combinations of Resistors
The 8.0- and 4.0resistors are in series and can be replaced with their equivalent, 12.0 The 6.0- and 3.0resistors are in parallel and can be replaced with their equivalent, 2.0 These equivalent resistances are in series and can be replaced with their equivalent resistance, 14.0
More examples
Direct Current and Alternating Current
When the current direction (not magnitude) in a circuit does not change with time, the current is called direct current
Most of the circuits analyzed will be assumed to be in steady state: with constant magnitude and direction, like the one powered through a battery.
When the current direction (often also the magnitude) in a circuit changes with time, the current is call alternating current.
The current from your car's alternator is AC.
Model of a battery
Two parameters, electromotive force (emf), , and the internal resistance r, are used to model a battery. When a battery is connected in a circuit, the electric potential measured at its + and terminals are called The terminal voltage V, with V = Ir If the internal resistance is zero, the terminal voltage equals the emf The internal resistance, r, does not change with external load resistance R, and this provides the way to measure the internal resistance.
V
battery V
load
PLAY ACTIVE FIGURE
Battery power figure
The power the battery generates (through chemical reactions):
p I (R r ) I 2
The power the battery delivers to the load, hence efficiency: pload V I R I2
efficiency = pload p R R r
battery V
load
The maximum power the battery can deliver to a load R pload 2 2 We have From pload R I and (R r ) I (R r )2
Where the emf is a constant once the battery is given.
From
dpload dR 1 (R r )
2
2R 2 (R r )3
0
We get R r to be the condition for maximum pload , or power delivered to the load.
Battery power figure
One can also obtain this result from the plot of
pload
Where when R battery V
2 (R r )2
r
R
pload reaches the maximum value
The efficiency of the battery at this point is 50% because
load
pload efficiency = p
R R r
More complicated circuits, circuits with 2+ batteries: Kirchhoff's Rules
A typical circuit that goes beyond simplifications with the parallel and series formulas: the current in the diagram. Kirchhoff's rules can be used to solve problems like this.
Rule 1: Kirchhoff's Junction Rule
Junction Rule, from charge conservation:
The sum of the currents at any junction must equal zero Mathematically:
I
junction
0
The example on the left figure:
I 1 - I2 - I3 = 0
Rule 2: Kirchhoff's Loop Rule
Choose your loop Loop Rule, from energy conservation:
The sum of the potential differences across all elements around any closed circuit loop must be zero Mathematically:
V1
V2
Loop direction
V
closed loop
0
Remember two things: 1. A battery supplies power. One needs to pay attention the Potential rises from the "" sign (+ or -) of these potential terminal to "+" terminal. changes, following the chosen loop 2. Current follows the direction of direction. electric field, hence the decrease of potential.
Kirchhoff's rules Strict steps in solving a problem
Step 1: choose and mark the loop. Step 2: choose and mark current directions. Mark the potential change on resistors. Step 3: apply junction rule:
I1
I2
L1
+
I3
+
L2
+
I1 I2 I3
0
Step 4: apply loop rule:
L1: +2.00I3 12.0 4.00I2 L2:
0
8.00 2.00I3 6.00I1 0
Step 5: solve the three equations for the three variables.
RC Circuits, solve with Kirchhoff's rules
When a circuit contains a resistor and a capacitor connected in series, the circuit is called a RC circuit. Current in RC circuit is DC, but the current magnitude changes with time. There are two cases: charging (b) and discharging (c).
Not a circuit
charging
Discharging
Charging a Capacitor
When the switch turns to position a, current starts to flow and the capacitor starts to charge.
+ Loop +
Kirchhoff's rule says:
Vc VR 0
Re-write the equation in terms of the charge q in C and the current I, and then only the variable q: q q dq RI 0 and then R 0 C C dt Solve for q: The current I is
q t
C 1 e
t RC
I t
dq dt
t RC e R
Here RC has the unit of time t, and is called the time constant.
Charging a Capacitor, graphic presentation
The charge on the capacitor varies with time
q(t) = C (1 e-t/RC) = Q(1 e-t/RC)
The current decrease with time
I( t )
e R
t RC
is the time constant = RC
Discharging a Capacitor
When the switch turns to position b, after the capacitor is fully charged to Q, current starts to flow and the capacitor starts to discharge.
+ Loop +
Kirchhoff's rule says:
Vc VR 0
Re-write the equation in terms of the charge q in C and the current I, and then only the variable q: q q dq RI 0 and then R 0 C C dt Solve for q: The current I is t t dq Q RC I t e q t Qe RC dt RC
Ii
Q RC
RC Circuit, Example
Adjust the values of R and C Observe the effect on the charging and discharging of the capacitor
PLAY ACTIVE FIGURE