Chapter 26 – Direct Current Circuits
I. Resistor Combinations
A. Resistors connected in series
B. Resistors connected in parallel.
C. Examples:
1. Find the equivalent resistance between points a and b. Connect a 15 volt battery across the circuit (connect to points a and b). Find the current through each resistor and the voltage across each resistor.
2. Find the equivalent resistance between points a and b for the combination shown.
3. Find the equivalent resistance of the combination shown.
4. Electricity in the home.
How are house plugs connected?
Why do fuses "blow" or circuit breakers "trip?"
II. Circuit Elements - How are voltage rises (+V) and voltage drops (-V) related to current directions for the circuit elements?
A. Capacitor: VC =
B. Resistor: VR = IR
C. Battery ( emf e and internal resistance r )
1. Supplying energy
2. Charging
D. Wires generally have small resistances that can be ignored compared to other elements in the circuit. In this case the wires act as equipotentials.
III. Circuit Analysis
A. Definitions:
1. Element in the circuit: resistor, capacitor, source, etc.
2. Junction: three or more elements joined together
3. Branch: the part of the circuit from one junction to the next
4. Loop: a closed path along which current can flow
5. Simple Loop: a loop with no branches physically enclosed within the loop
B. Kirchoff's Rules
1. Loop Theorem, = 0, the sum of the voltages around a closed loop equals zero.
= Vab + Vbc + Vcd + Vda = (Va – Vb) + (Vb – Vc) + (Vc – Vd) + (Vd – Va) = 0
2. Junction Theorem, = 0 or , the sum of the currents into a junction is zero or the sum of the magnitudes of the currents entering the junction equal the sum of the magnitudes of the currents leaving the junction.
IV. Examples
A. Single loop circuits
1. In the circuit shown, find
a. the current flow.
b. the power supplied by the battery.
c. the power dissipated in the resistor R.
2. In the circuit shown, find
a. the current flow.
b. the voltages Vab and Vbc .
(voltage divider)
3. In the circuit shown, find the current flow.
B. Multiple loop circuits using the branch-current method
1. In the circuit shown, find
a. the current in the branches
b. the voltage Vab .
2. Find the current in each branch and the voltage Vab .
3. Write the equations that will allow you to find the current in the branch a-b.
C. Multiple loop circuits using the loop-current method:
1. Repeat example 1 using the loop-current method.
2. Repeat example 2 using the loop-current method.
3. Repeat example 3 using the loop-current method.
V. The RC Series Circuit – transient effects, i.e., what happens immediately after the switch is connected to the battery?
A. Qualitative analysis
B. Quantitative analysis
1. At time t = 0, the switch is moved to position a.
2. At the time the current reaches the value Io , and the charge on the capacitor reaches a value of Qo , the switch is moved to position b. We choose to start the time again at t = 0.
3. Example: In an RC series circuit the resistance R = 1000 ohms, the capacitance C = 2 mF, and the emf e = 12 volts. Find the value of the voltage across the resistor VR and voltage across the capacitor VC at time t = 1 x 10-3 sec (1 msec).
Suppose the emf is disconnected from the circuit and the RC circuit is shorted. What are VR and VC after another 1 msec?
VI. The RC Series Circuit – initial and steady state situation, i.e., what happens when the switch is first closed and after the switch has been connected for a long time?
Example: In the circuit shown, the capacitor is initially uncharged. The switch is closed at time t = 0. Find the currents in the branches
A. immediately after the switch is closed.
B. after steady state has been reached (a relatively long time has elapsed).
VII. Electrical Measuring Instruments
A. Galvanometer – current detecting device. It consists of loops of wire wrapped around a core and placed between the poles of a magnet. When current flows through the wire loops, the loops undergo a deflection proportional to the current. The wires have resistance and therefore the galvanometer has resistance, RG .
B. Ammeter – current measuring device. It is placed in series with the elements in the branch where the current is to be found. The ideal ammeter should have no resistance in order not to affect the circuit.
In the circuit shown, the ammeter is measuring the current in the loop.
Example: Using a galvanometer with a resistance RG = 10 ohms and a sensitivity that gives a full-scale deflection for a current of 0.01 amps, construct a multirange ammeter that can measure currents from 0 to 0.5 amps and from 0 to 3 amps.
1. 0 to 0.5 amp range: What we want is for the ammeter to give a full-scale deflection when a current of 0.5 A flows through it. The ammeter is inserted at point P in the circuit. It consists of a "shunt" resistor and the galvanometer (RG).
The equivalent resistance of the ammeter can be found. This represents the amount of resistance that is placed in the circuit as a result of the ammeter being inserted.
2. 0 to 3 amp range: In the same way as above, an ammeter can be constructed that gives a full-scale deflection for a current of 3 A.
The equivalent resistance of this ammeter is found.
3. The effect of this ammeter in a circuit is shown. Connect a 6 volt battery to a 2 ohm resistor.
C. Voltmeter – voltage measuring device. It is placed in parallel with the element whose voltage is required or between the points where the voltage is to be found. The ideal voltmeter has an infinite resistance in order not to affect the circuit.
In the circuit shown, the voltmeter is measuring the voltage across the resistor.
Example: Using the same galvanometer as in the previous example, construct a voltmeter that measures voltages in the range from 0 to 1 volt and from 0 to 30 volts.
1. 0 to 1 volt range: We want the maximum voltage (1 volt in this case) to give a full-scale deflection.
The equivalent resistance of the voltmeter is found.
2. 0 to 30 volt range:
The equivalent resistance of the voltmeter is found.
3. The effect of the voltmeter is found for a circuit with a 50 volt battery connected to two resistors in series.
D. Ohmmeter – resistance measuring device
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