Electric Circuits and Resistance

Most of the electronic devices you use every day work because a flow of electric charges, called a current ( I in amperes A), gives energy to each of the components within the device. The energy is provided by an electric field which is set up by either a chemical reaction in a battery, or forces on charges by a magnetic field as explained by Faraday's law of electromagnetic induction. In a battery, the work that the charges can do as a result of traveling through a circuit from the positive terminal to the negative is described by the electric potential difference (V=Work/q), often referred to as the voltage of the battery. The electric potential difference is specifically the work per charge that the current is capable of providing to each part of an electric circuit and is measured in the SI unit volts (V).

Generally when a current travels through a conductor, the electrons flowing through the wire make collisions with the atoms in the metal. As a result, the movement of the charges which make up the current is slowed as energy is lost by the electrons when they move from one end of the wire to the other. The resistance of the conductor (R with SI units of ohms W) is used to calculate the losses due to the wire and depends on several factors including the materials used, operation temperature and the shape of the resistor. The voltage dropped across the resistor is found using Ohm's law V=IR which relates the current drawn by a battery through a single resistor to the voltage applied.

Many electrical devices are constructed out of a combination of resistances hooked up to a source of electric potential. In order to describe the characteristics of a circuit, it is common to calculate a total equivalent resistance for a complicated circuit in order to find the current drawn by a battery or the power provided. Examine the examples of circuit diagrams shown below. The symbol shown as short connected zig-zag lines is used to represent a resistor, and the symbol with the long and short parallel lines is a battery. For reference, the short line is the negative terminal of the battery, and the long line is the positive terminal. The battery provides a source of potential difference which gives the electric current the energy to do work in the circuit. In the figure to the left, the resistors are connected so that one runs into the next. This drawing shows a series combination of resistors and can be identified because the same current must go through each part of the circuit. The total equivalent resistance of a series combination is equal to the sum of each individual resistance : Rs=R1+R2+R3. Since the combined circuit can be described by a single resistance, the current can be obtained by Ohm's law I=V/Rs. The figure on the right shows a parallel combination of resistors. Here the resistances are connected so that the front of each resistor is attached to the front of the others, and the ends are also connected together. As the current flows for a parallel combination, there are multiple paths available for the current to travel so that the current is divided among the different circuit branches. In order to combine the resistances for a parallel combination, we must add the resistances as inverses 1/Rp=1/R1+1/R2+1/R3. Notice that the total equivalent resistance is actually the inverse of the equation given (you must once again flip the final sum). Once again, Ohm's law can be used to find the current characteristics of the circuit.

series                                                        parallel

The power (SI unit of Watts) which is provided by the battery can be calculated by using the current and equivalent resistance of the circuit. The power dissipated through a resistor is calculated from Joule's law as P=I2R. Many applications of electrical circuits directly use the energy produced by Joule heating. For example, the coils in a hair dryer or toaster oven use the flow of current to heat up the wires thereby warming the air around them. A light bulb glows because the resistance of the filament inside heats the metal to a high temperature until it radiates light energy. In each of these cases, the combined resistances of the circuit components can be treated using Ohm's law to describe the characteristics of the device.

Examples of electrical resistance

Example 1: A circuit is put together with a 4W, 7W, and 12W resistors connected in series across a 9 V battery. Calculate a) the total equivalent resistance for the circuit and b) the current drawn by the battery.

Solution: a) In order to calculate the total resistance for a series combination, you must simply add the resistances of each component: Rs=R1+R2+R3=4W+7W+ 12W = 23W.

b) The current is given by Ohms law: V=IR --> I= V/R = 9V/ 23W = 0.391 A.

Example 2: Three Christmas tree lights are hooked up in parallel across a 12 V battery. If the resistances of the light bulbs are 2W, 4W, and 6W, then what is a) the combined resistance of the circuit and b) the amount of current provided by the battery.

Solution: For resistances connected in parallel, the equivalent resistance is found by using the relation 1/Rp=1/R1+1/R2+1/R3=1/2W +1/4W +1/2W =0.916(W)-1. The resistance is obtained by taking one over this answer, or Rp=1 / 0.916(W)-1=1.09W.

b) Again the current is given by Ohms law: I= V/R = 12V/ 1.09W = 11 A

Example 3: A hair dryer with a resistance of 40W is connected to a 110 V outlet. Find a) the current through the hair dryer and b) the power used by the hair dryer.

Solution: a) Current is related to the resistance by Ohm's law as above I=V/R=110V/40W = 2.75A.

b) The power used by a resistance is calculated using Joule's law. The heat power then can be calculated by substituting into the formula P=I2R = (2.75A)2(40W)=302.5 W