Resistors Some substances are insulators. A battery will not make detectible current flow through them. Many substances (lead, iron, graphite, etc.) will let current flow. For most substances that are not insulators, the current is approximately proportional to the difference in voltage on the two sides. I = K(V C V D ) How much current flows depends on the substance, shape, and dimensions.
K in the equation I = K(V C V D ) depends on the substance, shape, & dimensions. The value of K for a material is called its conductance. The conductance is usually represented by the letter G. I = G(V C V D ) Notice again that it is the difference in voltages that determines the current. The reciprocal of the conductance is called the resistance. R = 1/G = (V C V D ) Memorize this A chunk of material with a wire on each side that is not an insulator is called a resistor. The bigger the resistance, the lower the conductance is. The bigger the resistance, the less it conducts. Notation for Voltage Differences Since current is determined by the difference of two voltages, we need a convenient notation for the difference of two voltages. The notation normally used is the following. V CD = V C V D V AB = V A V B etc. MEMORIZE THIS THOROUGHY!!
I = (V C V D ) = V CD = V CD = IR Ohm s Law Symbol for Resistor Wires Have Resistance Wires are usually made out of copper (Cu) or aluminum (Al). Cu & Al have resistance, but the resistance is very small. V A V B IR 0 since R is very small. (Remember what we said about wide pipes!) Since V A V B 0, V A V B. The voltage is approximately the same on both ends of a wire. A perfect wire is a wire with R = 0. We will work most problems as if wires are perfect.
Resistance Units I = IR = V AB R = The units of resistance are. This is called ohms. The resistance of a resistor in ohms is the number of volts across it needed to make one amp of current flow. A resistor which needs 5 volts to make one amp flow is called a 5 ohm resistor. R = = 5 ohms = 5 Ω. 1 ohm = 1 Ω. V = IR if V is in volts, I is in amps, & R is in ohms. 5 volts = (1 amp)(5 ohms) 10 volts = (2 amps)(5 ohms) 1 K Ω = 1000 Ω = 10 3 Ω 1 M Ω = 10 6 Ω
Current Units 1 Amp = 1 A = 1 1 ma = 1 milliamp = 10-3 Amp = 0.001 Amp 1 µa = 1 microamp = 10-6 Amp With I in amps & R in ohms, IR = volts across the resistor. If we put I in ma, it s 1000 times bigger. 7 Amps = 7000 ma. If we put R in KΩ, it s 1000 times smaller. 5 ohms = 0.005 KΩ. If we put I in ma & R in KΩ, then IR = volts still. I is 1000 times larger & R is a 1000 times smaller, so the product is the same as before. 5 Amps x 3 Ω = 15 volts 5000 ma x 0.003 KΩ = 15 volts Thus, V = IR with V in volts, I in ma, & R in KΩ.
Incompressibility The electron fluid in a wire or device is nearly incompressible. The number of electrons in a 1 mm 3 volume always approximately equals the number of protons. Suppose more electrons flowed into a region than flowed out, and the number of electrons in the region increased by 0.1%. Even this small percentage of extra electrons is a huge negative charge. The negative charge would repel electrons out of the region until the number of electrons once more equals the number of protons. The electrical forces in a conductor, or even a resistor, force the number of electrons in each region to remain almost exactly equal to the number of protons. Since the number of protons is fixed (protons don t move), this means that the number of electrons in each region never changes noticeably. The electron fluid is incompressible, like water in a pipe. For a wire, this implies that the current is the same at all points in the wire. If i A were different from i B, the number of electrons between A & B would be changing; it wouldn t be constant. Therefore i A = i B. Current is the same everywhere in a wire, even if the wire is interrupted by resistors or other devices.
The Main Cause of Errors in Circuit Analysis Most errors in circuit analysis are the result of not taking seriously enough one of the following two facts: 1. The current is the same everywhere in a wire. 2. The voltage is the same at all points connected by a wire. We will never find a case where either of these two facts is false, never. Keeping them in the front of your mind as you analyze circuits will be a tremendous help.