DC Circuits.

DC Circuits

EMF and Terminal Voltage
Resistors in Series and in Parallel Kirchhoff’s Rules EMFs in Series and in Parallel; Charging a Battery Circuits Containing Capacitors in Series and in Parallel

Units of Chapter 19 RC Circuits – Resistor and Capacitor in Series, parallel and combinations circuits Electric Hazards Ammeters and Voltmeters

The Electric Battery Volta discovered that electricity could be created if dissimilar metals were connected by a conductive solution called an electrolyte. This is a simple electric cell.

The Electric Battery A battery transforms chemical energy into electrical energy. Chemical reactions within the cell create a potential difference between the terminals by slowly dissolving them. This potential difference can be maintained even if a current is kept flowing, until one or the other terminal is completely dissolved.

The Electric Battery Several cells connected together make a battery, although now we refer to a single cell as a battery as well.

Electric Current Electric current is the rate of flow of charge through a conductor: Unit of electric current: the ampere, A. 1 A = 1 C/s.

A 2 mm long cross section of wire is isolated and 20 C of charge are determined to pass through it in 40 s. A 1 mm long cross section of wire is isolated and 2 C of charge are determined to pass through it in 0.5 s. I = _____ Ampere 0.5 4.0

Electric Current A complete circuit is one where current can flow all the way around. Note that the schematic drawing doesn’t look much like the physical circuit!

Electric Current In order for current to flow, there must be a path from one battery terminal, through the circuit, and back to the other battery terminal. Only one of these circuits will work:

Electric Current By convention, current is defined as flowing from + to -. Electrons actually flow in the opposite direction, but not all currents consist of electrons.

Microscopic View of Electric Current
Electrons in a conductor have large, random speeds just due to their temperature. When a potential difference is applied, the electrons also acquire an average drift velocity, which is generally considerably smaller than the thermal velocity.

Microscopic View of Electric Current
This drift speed is related to the current in the wire, and also to the number of electrons per unit volume.

There are two requirements which must be met in order to establish an electric circuit. There must be an energy supply capable doing work on charge to move it from a low energy location to a high energy location and thus establish an electric potential difference across the two ends of the external circuit. There must be a closed conducting loop in the external circuit which stretches from the high potential, positive terminal to the low potential, negative terminal.

Ohm’s Law: Resistance and Resistors
Experimentally, it is found that the current in a wire is proportional to the potential difference between its ends:

The ratio of voltage to current is called the resistance:

In many conductors, the resistance is independent of the voltage; this relationship is called Ohm’s law. Materials that do not follow Ohm’s law are called nonohmic. Unit of resistance: the ohm, Ω. 1 Ω = 1 V/A.

Ohm's Law: V = IR V = (6A)(3 Ω) = 18 volts (V)

Ohm's Law: I = V/R I = (12 V) /(3 Ω) = 4 amperes (A)

Ohm's Law: R = V/I R = (36 V) /(6 A) = 6Ω

Standard resistors are manufactured for use in electric circuits; they are color-coded to indicate their value and precision.

Some clarifications: Batteries maintain a (nearly) constant potential difference; the current varies. Resistance is a property of a material or device. Current is not a vector but it does have a direction. Current and charge do not get used up. Whatever charge goes in one end of a circuit comes out the other end.

True or False: The current at point E is considerably less than the current at point A since charge is being used up in the light bulbs. False

Resistivity The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area: The constant ρ, the resistivity, is characteristic of the material.

A cylindrical copper rod has resistance R
A cylindrical copper rod has resistance R. It is reformed to twice its original length with no change of volume. Its new resistance is: 1. R 2. 2R 3. 4R 4. 8R 5. R/2

Resistivity

Resistivity For any given material, the resistivity increases with temperature: Semiconductors are complex materials, and may have resistivities that decrease with temperature.

Electric circuit needs battery or generator to produce current – these are called sources of emf. Battery is a nearly constant voltage source, but does have a small internal resistance, which reduces the actual voltage from the ideal emf:

This resistance behaves as though it were in series with the emf.

What is the internal resistance of a 12
What is the internal resistance of a 12.0-V car battery whose terminal voltage drops to 8.4 V when the starter draws 75 A? What is the resistance of the starter?

Resistors in Series and in Parallel
A series connection has a single path from the battery, through each circuit element in turn, then back to the battery.

The current through each resistor is the same; the voltage depends on the resistance. The sum of the voltage drops across the resistors equals the battery voltage.

From this we get the equivalent resistance (that single resistance that gives the same current in the circuit).

Two cylinders are made of the same material and have the same length but different diameters. They are joined end-to-end and a potential difference is maintained across the combination. Which of the following quantities is the same for the two cylinders? 1. the potential difference 2. the current 3. the current density 4. none of the above

A parallel connection splits the current; the voltage across each resistor is the same:

The total current is the sum of the currents across each resistor:

This gives the reciprocal of the equivalent resistance:

An analogy using water may be helpful in visualizing parallel circuits:

4Ω 18Ω

Find the resistance between A and B:
1 / Rp = 1 / / / / 1 Rp = 0.50Ω VAB = 12 V What is the sum of all currents? What is the current in each resistor? 24A 2Ω 6A 3Ω, 4A 6Ω, 2A 1Ω, 12A

What is the total resistance between Points A and B?
4Ω + 6Ω = 10Ω 1 / Rp = 1 / / / 10 Rp = 2.98 Ω = 5.98Ω 1 / Rp = 1 / / 20 Rp = 4.60 Ω

What is the potential difference between points A and B?
1 / Rp = 1/3 + 1/6 Rp = 2 Ω Total resistance is 6 Ω I = 12 V / 6 Ω = 2 A Ohm's Law: VAB = (2 A) (2 Ω ) = 4 V

Three 45Ω lightbulbs and three 75Ω lightbulbs are connected in series
Three 45Ω lightbulbs and three 75Ω lightbulbs are connected in series. (a) What is the total resistance of the circuit? (b) What is their resistance if all six are wired in parallel? For the resistors in series the resistances add linearly. (b) For the resistors in parallel the resistances add reciprocally.

When hooked to identical batteries which circuit draws the most current?
a) Circuit A b) Circuit B c) Both the same

Kirchhoff’s Rules Some circuits cannot be broken down into series and parallel connections.

Kirchhoff’s Rules For these circuits we use Kirchhoff’s rules.
Junction rule: The sum of currents entering a junction equals the sum of the currents leaving it.

Kirchhoff’s Rules Loop rule: The sum of the changes in potential around a closed loop is zero.

Kirchhoff’s Rules Problem Solving: Kirchhoff’s Rules
Label each current. Identify unknowns. Apply junction and loop rules; you will need as many independent equations as there are unknowns. Solve the equations, being careful with signs.

EMFs in Series and in Parallel; Charging a Battery
EMFs in series in the same direction: total voltage is the sum of the separate voltages

EMFs in series, opposite direction: total voltage is the difference, but the lower-voltage battery is charged.

EMFs in parallel only make sense if the voltages are the same; this arrangement can produce more current than a single emf.

What happens when the wire is connected?

Connecting wire has no insulation and negligible resistance, are the birds safe to land where they did?

Loop BCDB: -4 I2 + 16 + 8 (I1 - I2) = 0 -4 I2 + 16 +8 I1 - 8 I2 = 0 8 I1 -12 I2 = -16
Loop ABDA: -2 I1 -8 (I 1- I2) = 0 -2 I1 -8 I 1 + 8 I = I1 + 8 I2 = 10 I1 = 1/7 A I2 = 10/7 A

Determine the magnitudes and directions of the currents in each resistor shown. The batteries have emfs of E1= 9V and E2=12V the resistors have values of R1=25Ω, R2= 18Ω and R3 =35Ω

There are three currents involved, and so there must be three independent equations to determine those three currents. One comes from Kirchhoff’s junction rule applied to the junction of the three branches on the right of the circuit.

Another equation comes from Kirchhoff’s loop rule applied to the top loop, starting at the negative terminal of the battery and progressing clockwise. The final equation comes from Kirchhoff’s loop rule applied to the bottom loop, starting at the negative terminal of the battery and progressing counterclockwise.

A little substitution

Electric Power Power, as in kinematics, is the energy transformed by a device per unit time:

Electric Power The unit of power is the watt, W.
For ohmic devices, we can make the substitutions:

P = I2R = (0.5)2 100 = 25 watts P = (V/R)2 R = V2/R = (50)2 /100 = 2500/100 = 25 watts P = I2(V/I) = IV = (0.50)50 = 25 watts

Determine the ... ... current in a 60-watt bulb plugged into a 120-volt outlet. b. ... current in a 120-watt bulb plugged into a 120-volt outlet. c. ... power of a saw that draws 12 amps of current when plugged into a 120-volt outlet. d. ... power of a toaster that draws 6 amps of current when plugged into a 120-volt outlet. 0.5 A 1 A 1440W 720W

A three-way lightbulb can produce 50 W, 100 W, or 150 W, at 120 V
A three-way lightbulb can produce 50 W, 100 W, or 150 W, at 120 V. Such a bulb contains two filaments that can be connected to the 120 V individually or in parallel. (a) Describe how the connections to the two filaments are made to give each of the three wattages. (b) What must be the resistance of each filament? A three way light has two filaments with different resistances.,the voltage is the same for each combination, the power and resistance are inversely related to each other. So for the 50 W output, use the higher-resistance filament . For the 100 W output, use the lower-resistance filament . For the 150 W output, use both of the filaments in parallel .

Electric Power What you pay for on your electric bill is not power, but energy – the power consumption multiplied by the time. We have been measuring energy in joules, but the electric company measures it in kilowatt-hours, kWh.

Which bulb has the greater filament resistance?
The 75 watt bulb has a greater resistance

If the four light bulbs in the circuits shown above are identical, which circuit puts out the most light?

10/10 RC circuits

Circuits Containing Capacitors in Series and in Parallel
Capacitors in parallel have the same voltage across each one:

In this case, the total capacitance is the sum:

Each capacitor sees same voltage: 100 V C = Q/V (basic definition)
Q = CV Q1 = (5)(100) = 500 C Q2 = (2)(100) = 200 C

Capacitors in series have the same charge:

In this case, the reciprocals of the capacitances add to give the reciprocal of the equivalent capacitance:

Six 47μf capacitors are connected in parallel
Six 47μf capacitors are connected in parallel. What is the equivalent capacitance? (b) What is their equivalent capacitance if connected in series?

CA = 50 x 10-6 F CB = 10 x 10-6 F What is the voltage on each capacitor? Q is the same on each capacitor. By Kirchhoff's Loop Rule: 12 - Q / CB - Q / CA = 0 Q = 1 x 10-4 C VA = Q / CA = 2 V VB = Q / CB = 10 V

1. (Ceq)a > (Ceq)b = (Ceq)c > (Ceq)d
Rank in order, from largest to smallest, the equivalent capacitance (Ceq)a to (Ceq)d of circuits a to d. 1. (Ceq)a > (Ceq)b = (Ceq)c > (Ceq)d 2. (Ceq)b > (Ceq)a = (Ceq)d > (Ceq)c 3. (Ceq)c > (Ceq)a = (Ceq)d > (Ceq)b 4. (Ceq)d > (Ceq)b = (Ceq)c > (Ceq)a 5. (Ceq)d > (Ceq)b > (Ceq)a > (Ceq)c STT30.5

RC Circuits – Resistor and Capacitor in Series
When the switch is closed, the capacitor will begin to charge.

The voltage across the capacitor increases with time: This is a type of exponential.

The charge follows a similar curve: This curve has a characteristic time constant:

The time constant for the discharge of this capacitor is
5. The capacitor doesn’t discharge because the resistors cancel each other. STT31.7

If an isolated charged capacitor is connected across a resistor, it discharges:

Ammeters and Voltmeters
An ammeter measures current; a voltmeter measures voltage. Both are based on galvanometers, unless they are digital. 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.

An ohmmeter measures resistance; it requires a battery to provide a current

If the meter has too much or (in this case) too little resistance, it can affect the measurement.

A galvanometer has an internal resistance of 30Ω and deflects full scale for a 50 μA current. Describe how to use this galvanometer to make (a) an ammeter to read currents up to 30 A, and (b) a voltmeter to give a full-scale deflection of 250 V. To make an ammeter, a shunt resistor must be placed in parallel with the galvanometer. The voltage across the shunt resistor must be the voltage across the galvanometer.

To make a voltmeter, a resistor must be placed in series with the galvanometer, so that the desired full-scale voltage corresponds to the full scale current of the galvanometer

10/10 RC circuits

An ammeter reads the current through a wire
10 Ω 10 V 10 Ω What does the ammeter read? A A A A 5. none of the above

In the movie Tango and Cash, Kurt Russell and Sylvester Stallone escape from a prison by jumping off the top of a tall wall through the air and onto a high-voltage power line. Before the jump, Stallone objects to the idea, telling Russell "We're going to fry." Russell responds with "You didn't take high school Physics did you. As long as you're only touching one wire and you're feet aren't touching the ground, you don't get electrocuted." Is this a correct statement? YES

Power in Household Circuits
The wires used in homes to carry electricity have very low resistance. However, if the current is high enough, the power will increase and the wires can become hot enough to start a fire. To avoid this, we use fuses or circuit breakers, which disconnect when the current goes above a predetermined value.

Electric Hazards Even very small currents – 10 to 100 mA can be dangerous, disrupting the nervous system. Larger currents may also cause burns. Household voltage can be lethal if you are wet and in good contact with the ground. Be careful!

Electric Hazards A person receiving a shock has become part of a complete circuit.

Electric Hazards Faulty wiring and improper grounding can be hazardous. Make sure electrical work is done by a professional.

Electric Hazards The safest plugs are those with three prongs; they have a separate ground line. Here is an example of household wiring – colors can vary, though! Be sure you know which is the hot wire before you do anything.

Fuses are one-use items – if they blow, the fuse is destroyed and must be replaced.

Circuit breakers, which are now much more common in homes than they once were, are switches that will open if the current is too high; they can then be reset.

Alternating Current Current from a battery flows steadily in one direction (direct current, DC). Current from a power plant varies sinusoidally (alternating current, AC).

Alternating Current The voltage varies sinusoidally with time:
as does the current:

Alternating Current Multiplying the current and the voltage gives the power:

Alternating Current Usually we are interested in the average power:

Alternating Current The current and voltage both have average values of zero, so we square them, take the average, then take the square root, yielding the root mean square (rms) value.

Superconductivity In general, resistivity decreases as temperature decreases. Some materials, however, have resistivity that falls abruptly to zero at a very low temperature, called the critical temperature, TC.

Superconductivity Experiments have shown that currents, once started, can flow through these materials for years without decreasing even without a potential difference. Critical temperatures are low; for many years no material was found to be superconducting above 23 K. More recently, novel materials have been found to be superconducting below 90 K, and work on higher temperature superconductors is continuing.

Summary Power in an electric circuit: Direct current is constant
Alternating current varies sinusoidally

Summary Ohmic materials have constant resistance, independent of voltage. Resistance is determined by shape and material: ρ is the resistivity.

Summary A battery is a source of constant potential difference.
Electric current is the rate of flow of electric charge. Conventional current is in the direction that positive charge would flow. Resistance is the ratio of voltage to current:

Summary of Chapter A source of emf transforms energy from some other form to electrical energy A battery is a source of emf in parallel with an internal resistance Resistors in series:

Summary of Chapter Resistors in parallel: Kirchhoff’s rules:
sum of currents entering a junction equals sum of currents leaving it total potential difference around closed loop is zero

Summary of Chapter Capacitors in parallel: Capacitors in series:

Summary of Chapter RC circuit has a characteristic time constant:
To avoid shocks, don’t allow your body to become part of a complete circuit Ammeter: measures current Voltmeter: measures voltage

Summary The average (rms) current and voltage:
Relation between drift speed and current:

Internet Archive: Details: Physics B Lesson 37: Capacitors in Circuits

DC Circuits.

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