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`physics.bu.edu/~duffy/PY 106/Resistancehtml
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`#287 captures
`8 Feb 1999-23 Jan 2
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`11h suvsalads dekalb idol BS ois ule dathan | whi i
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`Resistance and Ohm's Law
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`7-17-98
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`Direct current vs. alternating current
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`A battery produces direct current; the battery voltage (or emf) is constant, which generally results in a constant current flowing one way around a circuit. If the
`circuit has capacitors, which store charge, the current may not be constant, but it will still flowin one direction. The current that comes from a wall socket, on the
`other hand, is alternating current. With alternating current, the current continually changes direction. This is because the voltage (emf) is following a sine wave
`oscillation, changing from positive to negative and back again 60 times each second.
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`If you look at the voltage at its peak, it hits about +170 V, decreases through 0 to -170 V, and then rises back through 0 to +170 Vagain. (You might think this
`value of 170 Vshould really be 110 - 120 volts. That's actually a kind of average of the voltage, but the peak really is about 170 V_) This oscillating voltage
`produces an oscillating electric field; the electrons respond to this oscillating field and oscillate back and forth, producing an oscillating current in the circuit.
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`Electrical resistance
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`Voltage can be thought of as the pressure pushing charges along a conductor, while the electrical resistance of a conductor is a measure of howdifficult it is to
`push the charges along. Using the flowanalogy, electrical resistance is similar to friction. For water flowing through a pipe, a long narrowpipe provides more
`resistance to the flowthan does a short fat pipe. The same applies for flowing currents: long thin wires provide more resistance than do short thick wires.
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`The resistance (R) of a material depends on its length, cross-sectional area, and the resistivity (r ), a number that depends on the material:
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`R=plia
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`Resistance is measured in ohms, 2.
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`Theresistivity and conductivity are inversely related. Good conductors have low resistivity, while poor conductors (insulators) have resistivities that can be 20
`orders of magnitude larger.
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`Resistance also depends on temperature, usually increasing as the temperature increases. For reasonably small changes in temperature, the changein resistivity,
`and therefore the change in resistance, is proportional to the temperature change. This is reflected in the equations:
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`p=po[1+e(T- Tal]
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`and, equivalently,
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`A=RAo[1+0(T-Tol]
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`At lowtemperatures some materials, known as superconductors, have no resistance at all. Resistance in wires produces a loss of energy (usually in the form of
`heat), so materials with no resistance produce no energy loss when currents pass through them.
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`Ohm's Law
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`In many materials, the voltage and resistance are connected by Ohm's Law:
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`Ohm's Law : V=IR
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`The connection between voltage and resistance can be more complicated in some materials.These materials are called non-ohmic. We'll focus mainly on ohmic
`materials for now, those obeying Ohm's Law.
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`Example
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`A copper wire has a length of 160 m and a diameter of 1.00 mm. If the wire is connected to a 1.5-volt battery, how much current flows through the wire?
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`The current can be found from Ohm's Law, V =IR. The Vis the battery voltage, so if R can be determined then the current can be calculated. Thefirst step, then,
`is to find the resistance of the wire:
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`R=plia
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`Resistance is measured in ohms, 2.
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`Lis the length, 1.60 m. The resistivity can be found from the table on page 535 1n the textbook.
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`For copper, p= 1.72% 10% Om
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`The area is the cross-sectional area of the wire. This can be calculated using:
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`Aaa= (0.0005)? = 7.85 x 10-7 m2
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`The resistance of the wire is then:
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`Replids(1.72x 10%) (160) 1(7.95 x 107) = 3.500
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`The current can nowbe found from Ohm's Law:
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`I=V/R=1.5/3.5=0428A
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`Electric power
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`Power is the rate at which work is done. It has units of Watts. 1 W=1 I’s
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`Electric power is given by the equations:
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`P=¥I
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`P=¥2/R
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`P=12R
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`The power supplied to a circuit by a battery is calculated using P= VL
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`Batteries and power supplies supply power to a circuit, and this power 1s used up by motors as well as by anything that has resistance. The power dissipated ina
`resistor goes into heating the resistor; this is knowas Joule heating. In many cases, Joule heating is wasted energy. In some cases, however, Joule heating 1s
`exploited as a source of heat, such as in a toaster or an electric heater.
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`The electric company bills not for power but for energy, using units of kilowatt-hours.
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`TkWh= 3.6% 108 J
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`One kKW-h typically costs about 10 cents, which is really quite cheap. It does add up, though. The following equation gives the total cost of operating something
`electrical:
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`Cost = (Power rating in kW) x (numberof hours it's running) x (cost per kW-h)
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`An example...ifa 100 W light bulb is on for two hours each day, and energy costs $0.10 per kKW-h, how much doesit cost to run the bulb for a month?
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`Cost = 0.1 kW x 60 hours x $0.1/kW-h = $0.6, or 60 cents.
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`Try this at home- figure out the monthly cost of using a particular appliance you use every day. Possibilities include hair dryers, microwaves, TV's, etc. The power
`rating of an appliance like a TVis usually written on the back, and if it doesn't give the power it should give the current. Anything you plug into a wall socket runs
`at 120 V, so if you knowthat and the current you can figure out how much power it uses.
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`The cost for power that comes from a wall socket is relatively cheap. On the other hand, the cost of battery power is much higher. $100 per KW-h, a thousand times
`more than whatit costs for AC power from the wall socket, is a typical value.
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`Although power is cheap, it is not limitless. Electricity use continues to increase, so it is important to use energy more efficiently to offset consumption.
`Appliances that use energy most efficiently sometimes cost more but in the long run, when the energy savings are accounted for, they can end up being the cheaper
`alternative.
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`Series circuits
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`A series circuit is a circuit in which resistors are arranged in a chain, so the current has only one path to take. The current is the same through each resistor. The
`total resistance of the circuit is found by simply adding up the resistance values of the individual resistors:
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`equivalentresistance of resistors in series :A=Ay+Aa+ Ag+...
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`Ay
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`Aa
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`Fis
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`I
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`A series circuit is shown in the diagram above. The current flows through each resistor in turn. Ifthe values of the three resistors are:
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`Ay=Ge:, Aa=8O, andAs=40,
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`the totalresistance is 8+8+4=5 200.
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`With a 10 Vbattery, by V =I1R the total current in the circuit is:
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`T=V/R=10/20=0.5 A. The current through each resistor would be 0.5 A.
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`Parallel circuits
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`A parallel circuit is a circuit in which the resistors are arranged with their heads connected together, and their tails connected together. The current in a parallel
`circuit breaks up, with some flowing along each parallel branch and re-combining when the branches meet again. The voltage across each resistor in parallel is the
`same.
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`Thetotal resistance of a set of resistors in parallel is found by adding up the reciprocals of the resistance values, and then taking the reciprocal of the total:
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`equivalentresistance of resisters in parallel: 1#A=1/A,+1/Aa+11A3 +...
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`the resistance. If the values of the three resistors are:
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`Ry, =80, Ra=8O, and A3=4,
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`the total resistance is found by:
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`TiR=1/8411/8+114=112. Thisgves R=29.
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`With a 10 Vbattery, by V =IR the total current in the circuit is: T=V/R=10/2=3A.
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`The individual currents can also be found using I= V/ R. The voltage across each resistor is 10 V_ so:
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`Ty=10/8=17.254
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`Ta=10/8=1.254
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`Tse10/i4s254
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`Note that the currents add together to 5A, the total current.
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`A parallel resistor short-cut
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`If the resistors in parallel are identical, it can be very easy to work out the equivalent resistance. In this case the equivalent resistance of N identical resistors is the
`resistance of one resistor divided by N, the number of resistors. So, two 40-ohm resistors in parallel are equivalent to one 20-chm resistor; five 50-ohm resistors in
`parallel are equivalent to one 10-ohm resistor, etc.
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`Circuits with series and parallel components
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`Many circuits have a combination of series and parallel resistors. Generally, the total resistance in a circuit like this 1s found by reducing the different series and
`parallel combinations step-by-step to end up with a single equivalent resistance for the circuit. This allows the current to be determined easily. The current flowing
`through each resistor can then be found by undoing the reduction process.
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`General rules for doing the reduction process include:
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`1. Two (or more) resistors with thei heads directly connected together and their tails directly connected together are im parallel, and they can be reduced to one
`resistor using the equivalent resistance equation for resistors in parallel.
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`2. Two resistors connected together so that the tail of one is connected to the head ofthe next, with no other path for the current to take along the line
`connecting them, are in series and can be reduced to one equivalent resistor.
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`Finally, remember that for resistors in series, the current is the same for each resistor, and for resistors in parallel, the voltage is the same for each one.
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`Back to the lecture summary home page
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