Phys 151 F00-01 Lecture 14
Why are birds safe when they rest with both feet on transmission wires?
Overview
Electric Current: Going with the Flow
Resistance, not rebellion!
Electrical Power
The Electrical Circuit
Electric Current: Going with the Flow
Recall from Coulomb’s Law that there is a force between charges. If these charges are not held in place, they will move. With motion, work can be done. And in analogy to objects moving in the Earth’s gravitational field, where we say that a falling object loses potential energy and is then able to do work, the motion of a charged particle or body can be attributed to its moving from a place of high electric potential to a place of low electric potential (unless the charge is negative, in which case it is just the opposite).
When a charge Q of +1 C is able to do 1 J of work during its motion we say that it
has fallen through a potential difference DV of 1 volt.
Numerical exercise: What is the work done when a charge of +3 C drops through a potential difference of 2 volts?
Answer: 6 J
(from
W = Q.DV whereas in mechanics we have W = mgDh)
(In these terms, it can be seen that charge (in electrostatics) is equivalent to mass in mechanics while the potential difference DV plays the same role as gDh.)
Charges in motion
When an electric field is present, charges are forced to move. Such fields exist near charges. There are devices such as a battery which impose an electric field inside a conductor (such as a metallic wire; this is like putting a tilt to a flat surface). The battery also acts as a source of charge so a continuous stream of charges can be made to flow through the conductor if the battery is “applied” across it. We call this stream of charges an electric current. This is analogous to a tilted conveyor belt dispensing bottles or cans in a factory.
For a wire connected across a battery, the magnitude of the “current” (I) in it is defined as the rate of flow of charge through it. Thus if Q coulombs of charge flow past a point in the wire in t seconds, the current through that point is given by the formula
I = Q/t
The unit of current is the amp (ampere) and is the current when 1 C of charge flows by in 1 second.
Numerical exercise: If there is a current of 1 amp in a wire, how many coulombs of charge will flow by in 1 hour?
Answer: Q = I t = 1 x 3600 = 3600 coulombs
Resistance (is futile)
A current flows in a wire because charges are in continuous motion in it in a given direction (Some devices provide an oscillating potential difference so that the direction of flow of charge in the wire will alternate). These charges move because they are forced to do so by an electric field. The electric field can be pictured through a potential difference between points. And just as a steeper hill yields a faster moving sled down its slope, a larger potential difference between the two ends of a wire gives a larger current. This relationship is expressed through Ohm’s Law
DV = I R
where R is called the resistance of the wire. If DV is in volts and I is in amps, then R is measured in ohms. From Ohm’s Law, we see that for a given potential difference, a larger current flows if the resistance is smaller.
Obviously what R is depends upon the physical features of the wire and the material it is made of. As with water flow in a pipe, the “fatter” the wire the easier will charges flow through it. Current I should then increase, implying R is decreased. R is therefore inversely proportional to the cross-sectional area A of the wire.
For a given DV, if the length l of the wire is increased, the smaller is the potential gradient and the smaller is I.
Thus R is proportional to length l. Finally, the kind of material used in the wire plays a
role in deciding how quickly the charges are able to move through the wire. This feature is represented by the resistivity b of the wire. When all these factors are collected together, it is clear that the resistance of any length of wire can be given by
R = b l / A
Numerical exercise: The length of the wire connecting the terminals of a battery is shortened to half its original length. How is the resistance of the wire changed?
Answer: R is also halved.
Electrical Power
Since work is done in moving the charges along a wire, how do we find out what that work is? Since DV measures the potential difference between the two ends and if we assume that Q coulombs move in t seconds through the wire, the work done is
W = Q.DV
so that the power P expended is
P = W/t = DV.Q/t = I.DV = I2R = DV2/R
If I is in amps and V in volts, then P is in watts. The power does not just disappear when a current is running. It heats the wire.
Household appliances “draw” current (like a wire) in order to operate. They are designed to work on a particular voltage.
Numerical exercise: A 1100-watt hair dryer is designed to operate on 120 V. How much current does the dryer draw?
From P = I.DV
we have
I = P / DV
and since P = 1100 watts while DV = 120 volts the current is
I = 1100/120 = 9.2 amps
Numerical exercise: What is the cost of operating a 100 W lightbulb for 1 hr if the utility rate is $0.10 per kWhr?
The energy used in 1 hr is 100 Whr = 0.1 kWhr. This should cost 0.1 x 0.1 = $0.01.
The Electrical Circuit
When a current is able to flow through wires and appliances/devices the wires and appliances/devices are said to form an electrical circuit. The wires and appliances can be linked in two ways. If they are attached one after the other from one terminal of the battery (power source) to the other, this is called a series circuit. It is easy to see that this arrangement of “resistors” is tantamount to lengthening a wire. The overall resistance is then more than that of each one of the components. Indeed the resistance of the total combination is known to be the sum of the individual resistances. The current through each resistor is exactly the same.
A second arrangement of resistors is if every resistor is simultaneously connected to the two terminals of the battery or power source. This constitutes a parallel circuit and its effect is to create a combination resistor which is “fatter” than any of the component resistors. Thus the resistance of the total combination is decreased so that it is less than any one of the resistances in the circuit. The voltage across every resistor is the same.
Interesting Questions
Why are we told not to use electrical appliances in the bathroom?
Water is a relatively good conductor as is your body. If you touch the bare wire of an appliance which is plugged in, a circuit is created with a large drop in potential through your body to the “ground” (the nearest water outlet pipe); this produces significant current flow and disrupts the electrical signals which run all your important body functions.
Why do we jumpstart a car with a “dead” battery by connecting positive terminal to positive terminal and negative to negative and not vice versa?
We do so to create a parallel circuit and that minimizes the current flow through the dead battery and prevents damage to it.
Why are birds safe when they rest on transmission wires?
The body of the bird presents a large resistance and thus draws only a small current from the wires.
Why is electric power transmitted at high voltage?
This minimizes the percentage of energy loss through heating (I2 DV is much smaller than I2V).
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