Circuits Current Resistance & Ohm's Law Resistors in Series, in Parallel, and in combination Capacitors in Series and Parallel Voltmeters & Ammeters Resistivity Power & Power Lines Fuses & Breakers Bulbs in Series & Parallel
Electricity The term electricity can be used to refer to any of the properties that particles, like protons and electrons, have as a result of their charge. Typically, though, electricity refers to electrical current as a source of power. Whenever valence electrons move in a wire, current flows, by definition, in the opposite direction. As the electrons move, their electric potential energy can be converted to other forms like light, heat, and sound. The source of this energy can be a battery, generator, solar cell, or power plant.
Current By definition, current is the rate of flow of positive charge. Mathematically, current is given by: If 15 C of charge flow past some point in a circuit over a period of 3 s, then the current at that point is 5 C/s. A coulomb per second is also called an ampere and its symbol is A. So, the current is 5 A. We might say, "There is a 5 amp current in this wire." It is current that can kill a someone who is electrocuted. A sign reading "Beware, High Voltage!" is really a warning that there is a potential difference high enough to produce a deadly current.
Charge Carriers & Current A charge carrier is any charged particle capable of moving. They are usually ions or subatomic particles. A stream of protons, for example, heading toward Earth fronm the sun (in the solar wind) is a current and the protons are the charge carriers. Irn this case the current is in the direction of motion of protons, since protons are positively charged. In a wire on Earth, the charge carriers are electrons, and the current is in the opposite direction of the electrons. Negative charge moving to the left is equivalent to positive charge moving to the right. The size of the current depends on how much charge each carrier possesses, how quickly the carriers are moving, and the number of carriers passing by per unit time. protons Wire electrons
A circuit is a path through which an electricity carn flow. It often consists of a wire made of a highly conductive metal like copper. The circuit shown consists of a batteryf ), a resistor( and lengths of wire ( source of energy for the circuit. The potential difference across the battery is V. Valence electrons have a clockwise motion, opposite the direction of the current, I. The resistor is a circuit component that dissipates the energy that the charges acquired from the battery, usually as heat. (A light bulb, for example, would act as a resistor.) The greater the resistance, R, of the resistor, the more it restricts the flow of current. A Simple Circuit ). The battery is the
Building Analogy To understand circuits, circuit components, current, energy transformations within a circuit, and devices used to make measurements in circuits, we will make an analogy to a building Continued. V
Building Analogy Correspondences Battery Elevator that only goes up and all the way to the top floor Voltage of battery Height of building Positive charge carriers People who move through the building en masse (as a large group) CurrentTraffic (number of people per unit time moving past some point in the building) wire w/ no internal resistance Hallway (with no slope) Wire w/ internal resistance Hallway sloping downward slightly Resistor stairway, ladder, fire pole, slide, etc. that only goes down Voltage drop across resistorLength of stairway Resistance of resistorNarrowness of stairway Ammeter Turnstile (measures traffic without slowing it down) Voltmeter Tape measure (for measuring changes in height)
Current and the Building Analogy In our analogy people correspond to positive charge carriers and a hallway corresponds to a wire. So, when a large group of people move together down a hallway, this is like charge carriers flowing through a wire. Traffic is the rate at which people are passing, say, a water fountain in the hall. Current is rate at which positive charge flows past some point in a wire. This is why traffic corresponds to current. Suppose you count 30 people passing by the fountain over a 5 s interval. The traffic rate is 6 people per second. This rate does not tell us how fast the people are moving. We don't know if the hall is crowded with slowly moving people or if the hall is relatively empty but the people are running. We only know how many go by per second. Similarly, in a circuit, a 6 A current could be due to many slow moving charges or fewer charges moving more quickly. The only thing for certain is that 6 coulombs of charge are passing by each second
Battery & Resistors and the Building Analogy Our up-only elevator will only take people to the top floor, where they have maximum potential and, thus, where they are at the maximum gravitational potential. The elevator "energizes" people, giving them potential energy. Likewise, a battery energizes positive charges. Think of a 10 V battery as an elevator that goes up 10 stories. The greater the voltage, the greater the difference in potential, and the higher the building. As reference points, let's choose the negative terminal of the battery to be at zero electric potential and the ground floor to be at zero gravitational potential. Continued.. top floor hallway: high Uarav ow o + charges people bottom floor hallway: zero Ugrav
Battery & Resistors and the Building (cont.) Current flows from the positive terminal of the battery, where + charges are at high potential, through the resistor where they give up their energy as heat, to the negative terminal of the battery, where they have zero potential energy. The battery then "lifts them back up" to a higher potential. The charges lose no energy moving the a length of wire (with no internal resistance). Similarly, people walk from the top floor where they are at a high potential, down the stairs, where their potential energy is converted to waste heat, to the bottom floor, where they have zero potential energy. The elevator them lifts them back up to a higher potential. The people lose no energy traveling down a (level) hallway top floor hallway: high Uarav + charges people 0 r bottom floor hallway: zero Uarav
Resistance Resistance is a measure of a resistors ability to resist the flow of current in a circuit. As a simplistic analogy, think of a battery as a water pump; it's voltage is the strength of the pump. A pipe with flowing water is like a wire with flowing current, and a partial clog in the pipe is like a resistor in the circuit. The more clogged the pipe is, the more resistance it puts up to the flow of water trying to flow through it, and the smaller that flow will be. Similarly, if a resistor has a high resistance, the current flowing it will be small. Resistance is defined mathematically by the equation: Resistance is the ratio of voltage to current. The current flowing through a resistor depends on the voltage drop across it and the resistance of the resistor. The SI unit for resistance is the ohm, and its symbol is capital omega: 2. An ohm is a volt per ampere: 12 1 V/A
Resistance and Building Analogy In our building analogy we're dealing with people instead of water molecules and staircases instead of clogs. A wide staircase allows many people to travel down it simultaneously, but a narrow staircase restricts the flow of people and reduces traffic. So, a resistor with low resistance is like a wide stairway, allowing a large current though it, and a resistor with high resistance is like a narrow stairway, allowing a smaller current. 1-2 A 1 4 A R 32 Narrow staircase means reduced traffic. Wide staircase means more traffic.
Ohmic vs. Nonohmic Resistors If Ohm's law were always true, then as V across a resistor increases, so would I through it, and their ratio, R (the slope of the graph) would remain constant In actuality, Ohm's law holds only for currents that aren't too large. When the current is small, not much heat is produced in a real, so resistance is constant and Ohm's law holds (linear portion of graph). But large currents cause R to increase (concave up part of graph) Ohmic Resistor Real Resistor
Resistors in Series: Building Analogy Elevator (battery) R2 R3 3 steps To go from the top to the bottom floor, all people must take the same path. So, by definition, the staircases are in series. With each flight people lose some of the potential energy given to them by the elevator, expending all of it by the time they reach the ground floor. So the sum of the V drops across the resistors the voltage of the battery. People lose more potential energy going down longer flights of stairs, so from V IR, long stairways correspond to high resistance resistors. The double waterfall is like a pair of resistors in series because there is only one route for the water to take. The longer the fall, the greater the resistance
Equivalent Resistance in Series If you were to remove all the resistors from a circuit and replace them with a single resistor, what resistance should this replacement have in order to produce the same current? This resistance is called the equivalent resistance, Rear In series Req is simply the sum of the resistances of all the resistors, no matter how many there are: eR R2 +R3+. Mnemonic: Resistors in Series are Really Simple R1 R2 R3 R, eq
Series Solution 1. Since the resistors are in series, simply add the three resistances to find Rea 4 eq 2. To find Io(the current through the battery), use VIFR: 6 12. So, 1 6/12 0.5 A 3. Since the current throughout a series circuit is constant, use V- IR with each resistor individually to find the V drop across each. Listed clockwise from top: V 6 (0.5)(4) 2 V V2 (0.5) (2) 1V V3 (0.5)(6) 3 V Note the voltage drops sum to 6 V.
Resistors in Parallel: Building Analogy Elevator (battery) R 2 Suppose there are two stairways to get from the top floor all the way to the bottom. By definition, then, the staircases are in parallel. People will lose the same amount of potential energy taking either, and that energy is equal to the energy the acquired from the elevator. So the Vdrop across each resistor equals that of the battery. Since there are two paths, the sum of the currents in each resistor equals the current through the battery. A wider staircase will accommodate more traffic, so from V= IA a wide staircase corresponds to a resistor with low resistance. The double waterfall is like a pair of resistors in parallel because there are two routes for the water to take. The wider the fall, the greater the flow of water, and lower the resistance.
Equivalent Resistance in Parallel 1,+ 12 + 13/ (currents in branches sum to current through battery) V-11 R1, V= 12 A, and V= 13 R (Vis a constant in parallel) (substitution) 3 eq (divide through by V) This formula extends to any number of resistors in parallel. eq eq
1st Color Band Multiplier Color Band Color Code for Resistors Color coding is a system of marking the resistance of a resistor. It consists of four different colored bands that are used to figure out the resistance in ohms. 2nd Color Band Color Band The first two bands correspond to a two-digit number. Each color corresponds to a particular digit that can looked up on a color chart . The third band is called the multiplier band. This is the power of ten to be multiplied by your two-digit number. The last band is called the tolerance band. It gives you an error range for the labeled resistance
Color Code Example A resistor color code has these color bands: Calculate its resistance and accuracy (yellow, green, red, gold) 1. Look up the corresponding numbers for the first three colors Yellow= 4, Green 5, Red=2 2. Combine the first two digits and use the multiplier: 45 x 102 4500 3. Find the tolerance corresponding to gold and calculate the maximum error: Gold-5% and 0.05(4500)-225. So, the resistance is 4500 225
Capacitor Review .As soon as switch S is closed a clock-wise current will flow, depositing positive charge on the right plate, leaving the left plate negative. This current starts out as V/R, but it decays to zero with time because as the charge on the capacitor grows the voltage drop across it grows too. As soon as Veap- V, the current ceases. The cap. maintains a charges separation, equal but opposite charges. When S is closed, Q increases from zero to C Vcap C is the capacitance of the capacitor, its charge storing capacity. The bigger C is, the more charge the cap. can store at a given voltage. At any point in time Q C V Even when removed from Q +Q the circuit, the cap. can maintain its charge separation and result in a shock A charged cap. stores electrical potential energy in an electric field between its plates. The battery stores chemical potential energy (chemical reactions supply charge carriers with potential energy). The resistor does not store energy; rather it dissipates energy as heat whenever current flows through it.
Parallel Capacitors If we removed all capacitors in a circuit and replaced them with a single capacitor, what capaciatance should it have in order to store the same charge as the original circuit? This is called the equivalent capacitance, Ceq. In 91 parallel the voltage drop across each resistor is the same, just as it was with resistors. Because the capacitances may differ, the charge on each capacitor may differ. From Q CV: 2 92 C2 The total charged stored is: total eq 1 + 2. In general Ce C1 C2+ Ca+ eq
Capacitors in Series In series the each capacitor holds the same charge, even if they have different capaci-tances. Here's why: The battery "rips off" a charge -q from the right side of C1 and deposits it on the left side of C3 Then the left side of C3 repels a charge -q from its right plate. over to the left side of C2. Meanwhile, the right side of C1 attracts a charge -q from the right side of C2. Charges don't jump across capacitors, so the green "H" and the blue "H" are isolated and must remain neutral. This forces all capacitors to have the same charge. The total charge is really just q, since this is the only charge acted on by the battery. The inner H's could be removed and it wouldn't make a difference. C, 9 2 total
Capacitors in Series (cont.) O, from C, 9 3 (since each the charge on each capacitor is the same as the total charge) This yields: 9total eq In general, for any number in parallel eq
Capacitor-Resistor Comparison V IR Resistors Capacitors Currents Voltages Series same add Parallel add same Charges Voltages Series same add Parallel add same Series: Req_ R Series: Parallel: Parallel: "Resistors in Series "Parallel Capacitors are are Really Simple." a Piece of Cake." he formulae for series are parallel are reversed simply because in the defining equations at the top, Ris replaced with 1/C
Power Recall that power is the rate at which work is done. It can also be defined as the rate at which energy is consumed or expended: energy time Power- For electricity, the power consumed by a resistor or generated by a battery is the product of the current flowing through the component and the voltage drop across it: P-I V Here's why: By definition, current is charge per unit time, and voltage is energy per unit charge. So, energy_ p energy charge time time
Power: SI Units the SI unit for power is the watt. By definition: 1W=1J/s A watt is equivalent to an ampere times a volt: This is true since (1 C/s) (1 J/C)-1 J/s 1 W.
Resistivitv & Resistance Resistance is an object property. It represents the degree to which an object resists flow of current. Resistivity is a material property. It represents the degree to which a material comprising an object resists flow of current. Ex: A wire is an object and it has some internal resistance. Copper is common material used to make wire and it has a known, small resistivity. The resistivity of copper is the same in any wire, but different wires have different internal resistances, depending on their lengths and diameters. A wire's resistance is proportional to its length (imagine every meter of wire with a tiny, built-in resistor) and inversely proportional to its cross-sectional area (just as a wider pipe allows greater flow of water). The constant of proportionality is the resistivity: R resistance of the wire p resistivity of the metal in wire L = length of the wire A cross sectional area of the wire
Resistivity: SI Units The SI unit for resistivity is an ohm-meter: 2m, as can be deduced from the formula: Copper has a resistivity of 1.69 x 10-8 2 m. The internal resistance of a copper wire depends on how long and how thick it is, but since p is so small, the resistance of the wire is usually negligible. Resistivity is considered a constant, at least at a given temperature. Resistivity increases slightly with temperature. This is why resistors behave in a nonohmic fashion when the current is high--high current leads to high temperatures, which increases resistivity, which increases resistance.
Power Lines Power is transmitted from power plants via power lines using very high voltages. Here's why: A certain amount of power must be supplied to a town. From P-I V, either current or voltage must high in order to meet the needs of a power hungry town. If the current is high, the power dissipated by the internal resistance of the long wires is significant, since this power is given by transformer P 2 R. Power companies use high voltage so that the current can be smaller. This minimizes power loss in the line. At your house voltage must be decreased significantly. This is accomplished by a transformer, which can step up or step down voltages.