**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
: R_{s}=R_{1}+R_{2}+R_{3}.
Since the combined circuit can be described by a single resistance, the current
can be obtained by Ohm's law I=V/R_{s}. 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/R_{p}=1/R_{1}+1/R_{2}+1/R_{3}.
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 ^{2}R. 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: R_{s}=R_{1}+R_{2}+R_{3}=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/R_{p}=1/R_{1}+1/R_{2}+1/R_{3}=1/2W +1/4W
+1/2W =0.916(W)^{-1}. The resistance is obtained by taking one over
this answer, or R_{p}=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=I^{2}R = (2.75A)^{2}(40W)=302.5 W