1. Direct Current Circuits
Chapter 18
Direct Current Circuits

2. 18.1 Sources of emf
The source that maintains the current in a closed circuit is called a source of emf
Emf=electromotive force
Any devices that increase the potential energy of charges circulating in circuits are sources of emf
Examples include batteries and generators

3. Emf and Internal Resistance
A real battery has some internal resistance
Therefore, the terminal voltage is not equal to the emf

4. More About Internal ResistanceV
The schematic shows the internal resistance, r
The terminal voltage is V = ε – Ir (source minus the internal loss)
For the entire circuit,
ε = IR + Ir =I (R+r)

5. Internal Resistance and emf, cont
ε is equal to the terminal voltage when the current is zero (open circuit)
Also called the open-circuit voltage
R is called the load resistance
The current depends on both the resistance external to the battery and the internal resistance

6. 18.2 Resistors in Series
When two or more resistors are connected end-to-end, they are said to be in series
The current is the same in resistors because any charge that flows through one resistor flows through the other
The sum of the potential differences across the resistors is equal to the total potential difference across the combination

7. Resistors in Series, cont
V1=IR1
V2=IR2
Voltages add
V =IR1 + IR2
V =I (R1+R2)
Req= R1+R2
The equivalent resistance Req has the same effect on the circuit as the original combination of resistors V

8. Equivalent Resistance – Series
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 of the individual resistance ( see parallel connection of capacitors!)

9. Equivalent Resistance – Series An Example
Four resistors are replaced with their equivalent resistance

10. 18.3 Resistors in Parallel
The potential difference across each resistor is the same because each is connected directly across the battery terminals
The current, I, that enters a point must be equal to the total current leaving that point
I = I1 + I2
The currents are generally not the same
Consequence of Conservation of Charge

11. Equivalent Resistance – Parallel, cont
I1=V/R1 and I2=V/R2. The complete current provided by the source is given by, I=I1+I2=V(1/R1+1/R2)=V/Req.
1/Req=1/R1+1/R2
Req is the equivalent resistance for a parallel circuit
Household circuits are wired so the electrical devices are connected in parallel
Circuit breakers may be used in series with other circuit elements for safety purposes

12. Equivalent Resistance – Parallel
The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistance ( see series connection of capacitors)
The equivalent is always less than the smallest resistor in the group

13. Problem-Solving Strategy, 1
When two or more resistors are connected in series, they carry the same current, but the potential differences across them are not the same.
The resistors add directly to give the equivalent resistance of the series combination

14. Problem-Solving Strategy, 2
When two or more resistors are connected in parallel, the potential differences across them are the same. The currents through them are not the same.
The equivalent resistance of a parallel combination is found through reciprocal addition
The equivalent resistance is always less than the smallest individual resistor in the combination

15. Problem-Solving Strategy, 3
A complicated circuit consisting of several resistors and batteries can often be reduced to a simple circuit with only one resistor
Replace any resistors in series or in parallel using steps 1 or 2.
Sketch the new circuit after these changes have been made
Continue to replace any series or parallel combinations
Continue until one equivalent resistance is found

16. Problem-Solving Strategy, 4
If the current in or the potential difference across a resistor in a complicated circuit is to be identified, start with the final circuit found in step 3 and gradually work back through the circuits
Use V = I R and the procedures in steps 1 and 2

17. Example Compute the equivalent resistance and the current of the network (a) below.

18. Solution Step 1: 1/(3 W)+1/(6 W)=1/Rp  Rp=2 W (b)
Rp=Req for parallel connection and Rs=Req for series connection
Step 1: 1/(3 W)+1/(6 W)=1/Rp  Rp=2 W (b)
Step 2: 4 W+2 W=Rs  Rs=6 W (c)
Step 3: I=V/Rs=18 V/6 W=3 A (d)
Step 4: V=(3 A)(4 W+2 W)=12 V+6 V=18 V (e)
Step 5: 6 V/6 W=1 A and 6 V/3 W=2 A (f)

19. Shortcut
For n parallel equal resistors, the equivalent Rp is expressed by  
Rp=R1/n
The same shortcut is valid for n equal capacitors in series 
Cs=C1/n

20. 18.4 Kirchhoff’s Rules
There are ways in which resistors and batteries can be connected so that the circuits formed cannot be reduced to a single equivalent resistor (two examples are shown on the right)
Two rules, called Kirchhoff’s Rules can be used instead

21. Statement of Kirchhoff’s Rules
Junction Rule ( I = 0)
The sum of the currents entering any junction must equal the sum of the currents leaving that junction
A statement of Conservation of Charge
Loop Rule ( U = 0)
The sum of the potential differences across all the elements around any closed circuit loop must be zero
A statement of Conservation of Energy

22. More About the Junction Rule
I1 = I2 + I3
From Conservation of Charge
Diagram (b) shows a mechanical analog

23. More About the Loop Rule
The voltage across a battery is taken to be positive (a voltage rise) if traversed from – to + and and negative if traversed in the opposite direction.
The voltage across a resistor is taken to be negative (a drop) if the loop is traversed in in the direction of the assigned current and positive if traversed in the opposite direction

24. Setting Up Kirchhoff’s Rules
Assign symbols and directions to the currents in all branches of the circuit
If a direction is chosen incorrectly, the resulting answer will be negative, but the magnitude will be correct
When applying the loop rule, choose a direction for transversing the loop
Record voltage drops and rises as they occur

25. Problem-Solving Strategy – Kirchhoff’s Rules
Draw the circuit diagram and assign labels and symbols to all known and unknown quantities. Assign directions to the currents.
Apply the junction rule to any junction in the circuit
Apply the loop rule to as many loops as are needed to solve for the unknowns
Solve the equations simultaneously for the unknown quantities.

26. Example: Find the current I, r and e.
Junction rule at a:
2 A+1 A-I=0
I=3 A
Loop (1):
12 V-Ir -(3 W)(2 A)=0
r=12 V/(3 A)-6 V/(3 A)
r=2 W
Loop (2):
-e+(1 W)(1 A)-( 3 W)(2 A)=0
e=-5 V (polarity is opposite!)
Check with loop (3):
12 V-(2 W)(3 A)-(1 W)(1 A)+e =0
e=-5 V