1. Chapter 26 DC Circuits
Chapter 26 Opener. These MP3 players contain circuits that are dc, at least in part. (The audio signal is ac.) The circuit diagram below shows a possible amplifier circuit for each stereo channel. Although the large triangle is an amplifier chip containing transistors (discussed in Chapter 40), the other circuit elements are ones we have met, resistors and capacitors, and we discuss them in circuits in this Chapter. We also discuss voltmeters and ammeters, and how they are built and used to make measurements.

2. 26-4 Series and Parallel EMFs; Battery Charging
EMFs in series in the same direction: total voltage is the sum of the separate voltages.

3. 26-4 Series and Parallel EMFs; Battery Charging
EMFs in series, opposite direction: total voltage is the difference, but the lower-voltage battery is charged. Example :”car alternator”

4. 26-4 Series and Parallel EMFs; Battery Charging
EMFs in parallel only make sense if the voltages are the same; this arrangement can produce more current than a single emf. This arrangement is not recommended, since there is a loss in internal resistance.

5. 26-5 Circuits Containing Resistor and Capacitor (RC Circuits)
When the switch is closed, the capacitor will begin to charge. As it does, the voltage across it increases, and the current through the resistor decreases.
Figure After the switch S closes in the RC circuit shown in (a), the voltage across the capacitor increases with time as shown in (b), and the current through the resistor decreases with time as shown in (c).

6. 26-5 Circuits Containing Resistor and Capacitor (RC Circuits)
To find the voltage as a function of time, we write the equation for the voltage changes around the loop:
Since I = dQ/dt, we can integrate to find the charge as a function of time:

7. 26-5 Circuits Containing Resistor and Capacitor (RC Circuits)
The voltage across the capacitor is VC = Q/C:
The quantity RC that appears in the exponent is called the time constant of the circuit:

8. 26-5 Circuits Containing Resistor and Capacitor (RC Circuits)
The current at any time t can be found by differentiating the charge:
is the time constant
of the RC circuit
τ = RC

9. 26-5 Circuits Containing Resistor and Capacitor (RC Circuits)
Example 26-11: RC circuit, with emf.
The capacitance in the circuit shown is C = 0.30 μF, the total resistance is 20 kΩ, and the battery emf is 12 V. Determine (a) the time constant, (b) the maximum charge the capacitor could acquire, (c) the time it takes for the charge to reach 99% of this value, (d) the current I when the charge Q is half its maximum value, (e) the maximum current, and (f) the charge Q when the current I is 0.20 its maximum value.
Solution: a. The time constant is RC = 6.0 x 10-3 s.
b. The maximum charge is the emf x C = 3.6 μC.
c. Set Q(t) = 0.99 Qmax and solve for t: t = 28 ms.
d. When Q = 1.8 μC, I = 300 μA.
e. The maximum current is the emf/R = 600 μA.
f. When I = 120 μA, Q = 2.9 μC.

10. 26-5 Circuits Containing Resistor and Capacitor (RC Circuits)
If an isolated charged capacitor is connected across a resistor, it discharges:
Figure For the RC circuit shown in (a), the voltage VC across the capacitor decreases with time, as shown in (b), after the switch S is closed at t = 0. The charge on the capacitor follows the same curve since VC α Q.
Q0 is the maximum charge

11. 26-5 Circuits Containing Resistor and Capacitor (RC Circuits)
Once again, the voltage and current as a function of time can be found from the charge:
and

12. Problem 48
48.(II) The RC circuit of Fig. 26–59 (same as Fig. 26–18a) has R=8.7kΩ and C=3.0µF The capacitor is at voltage V0 at t=0s when the switch is closed. How long does it take the capacitor to discharge to 0.10% of its initial voltage?

13. 26-5 Circuits Containing Resistor and Capacitor (RC Circuits)
Example 26-12: Discharging RC circuit.
In the RC circuit shown, the battery has fully charged the capacitor, so Q0 = C E. Then at t = 0 the switch is thrown from position a to b. The battery emf is 20.0 V, and the capacitance C = 1.02 μF. The current I is observed to decrease to 0.50 of its initial value in 40 μs. (a) What is the value of Q, the charge on the capacitor, at t = 0? (b) What is the value of R at 40 μs? (c) What is Q at t = 60 μs?
Solution: a. At t = 0, Q = CE = 20.4 μC.
b. At t = 40 μs, I = 0.5 I0. Substituting in the exponential decay equation gives R = 57 Ω.
c. Q = 7.3 μC.

14. 26-7 Ammeters and Voltmeters
An ammeter measures current; a voltmeter measures voltage. Both are based on galvanometers, unless they are digital.
Figure An ammeter is a galvanometer in parallel with a (shunt) resistor with low resistance, Rsh.

15. Ammeters and votmeters
The current in a circuit passes through the ammeter; the ammeter should have low resistance so as not to affect the current.
A voltmeter should not affect the voltage across the circuit element it is measuring; therefore its resistance should be very large.

16. 26-7 Ammeters and Voltmeters
Summary: How to connect Meters?
An ammeter must be in series with the current it is to measure;
A voltmeter must be in parallel with the voltage it is to measure.

17. Chapter 27 Magnetism
Chapter 27 opener. Magnets produce magnetic fields, but so do electric currents. An electric current flowing in this straight wire produces a magnetic field which causes the tiny pieces of iron (iron “filings”) to align in the field. We shall see in this Chapter how magnetic field is defined, and that the magnetic field direction is along the iron filings. The magnetic field lines due to the electric current in this long wire are in the shape of circles around the wire. We also discuss how magnetic fields exert forces on electric currents and on charged particles, as well as useful applications of the interaction between magnetic fields and electric currents and moving electric charges.

18. 27-1 Magnets and Magnetic Fields
Magnets have two ends – poles – called north and south.
Like poles repel; unlike poles attract.
Figure Like poles of a magnet repel; unlike poles attract. Red arrows indicate force direction.

20. 27-1 Magnets and Magnetic Fields
However, if you cut a magnet in half, you don’t get a north pole and a south pole – you get two smaller magnets.
Figure If you split a magnet, you won’t get isolated north and south poles; instead, two new magnets are produced, each with a north and a south pole.

21. 27-1 Magnets and Magnetic Fields
Magnetic fields can be visualized using magnetic field lines, which are always closed loops.
Figure (a) Visualizing magnetic field lines around a bar magnet, using iron filings and compass needles. The red end of the bar magnet is its north pole. The N pole of a nearby compass needle points away from the north pole of the magnet. (b) Magnetic field lines for a bar magnet.

22. Similarities with Electric Fields
Magnetic Fields
Similarities with Electric Fields
Electric
Magnetic
North and South poles
Like poles repel
Opposite poles attract
Field lines outside the material move from N to S
Positive and Negative Charges
Like Charges repel
Opposite Charges attract
Field lines move from + to -

23. Magnetic Fields Similarities with Electric Fields Electric Field
•tangent to the field lines
•the strongest where the field lines are the closest
•tangent to the field lines
•the strongest where the field lines are the closest

24. 27-1 Magnets and Magnetic Fields
The Earth’s magnetic field is similar to that of a bar magnet.
Note that the Earth’s “North Pole” is really a south magnetic pole, as the north ends of magnets are attracted to it.
Figure The Earth acts like a huge magnet; but its magnetic poles are not at the geographic poles, which are on the Earth’s rotation axis.
Canadian Artic

25. 27-1 Magnets and Magnetic Fields
A uniform magnetic field is constant in magnitude and direction.
The field between these two wide poles is nearly uniform.
Figure Magnetic field between two wide poles of a magnet is nearly uniform, except near the edges.