**Unit 13: Electric Circuits**

**Problem Set A**

**Overview:**

Problem Set A targets your understanding of the mathematics of basic electric circuit relationships. There are 20 problems in the set which test your ability to manipulate the following mathematical equations.

**Electric Current**

When charge flows through the wires of an electric *circuit*,
current is said to exist in the wires. Electric current is actually a
quantifiable notion which is defined as the rate at which charge
flows past a point on the circuit. It can be determined by simply
measuring the quantity of charge that flows past a cross-sectional
area of a wire on the circuit. As a rate quantity, current
(**I**) is expressed by the following
equation

where **Q** is the quantity of
charge flowing by a point in a time period of
**t**. The standard metric unit for
the quantity current is the ampere, often abbreviated as Amps or A.
One ampere is equivalent to a Coulomb of charge flowing past a point
in 1 second. Since the quantity of charge passing a point on a
circuit is related to the number of mobile charge carriers
(electrons) which move past that point, the current can also be
related to the number of electrons and the time. This relationship
can be made using the quantity of charge on a single electron.

**Resistance**

As charge flows through a circuit, it encounters *resistance*
or a hindrance to its flow. Like current, resistance if a
quantifiable term. The quantity of resistance offered by a section of
wire depends upon three variables - the material the wire is made out
of, the length of the wire, and the cross-sectional area of the wire.
One physical property of a material is its resistivity - a measure of
that materials tendency to resist charge flow through it. Resistivity
values for various conducting materials are typically listed in
textbooks. Knowing the material the wire is composed of and its
length (**L**) and cross-sectional
area (**A**), the resistivity value
(), its
resistance (**R**) can be determine
using the equation below.

The standard metric unit of resistance is the ohm (abbreviated by the greek letter ).

The main difficulty with the use of the above equation pertains to
the units of expression of the various quantities. The resistivity
() is typically
expressed in ohm•m. Thus, the length should be expressed in
units of m and the cross-sectional area in m^{2}. Often times
the radius or diameter of the a wire with a circular cross section is
stated and the area must be determined using the formula for the area
of a circle.

**Ohm's Law Equation**

The amount of current that flows in a circuit is dependent upon
two variables. Current is inversely proportional to the overall
resistance (**R**) of the elements
within the circuit and directly proportional to the electric
potential difference impressed across the circuit. The electric
potential difference (**V**)
impressed across a circuit is simply the voltage supplied by the
energy source (batteries, outlets, etc.). For homes in the United
States, this value is close to 120 Volts. The mathematical
relationship between current, voltage and resistance is expressed by
the following equation

**Power**

Electrical circuits are all about energy. Energy is put into a
circuit by the battery or the commerical electricity supplier. The
elements of the circuit (lights, heaters, motors, refrigerators, and
even wires) convert this electric potential energy into other froms
of energy such as light energy, sound energy, thermal energy and
mechanical energy. **Power** refers
to the rate at which energy is supplied or converted by the appliance
or circuit. It is the rate at which energy is lost or gained at any
given location within the circuit. As such, the generic equation for
power is

The energy loss (or gain) is simply the product of the electric potential difference between two points and the quantity of charge which moves between those two points in a time period of t. As such, the energy loss (or gain) is simply V • Q. When this expression is substituted into the above equation, the power equation becomes

By combining the Ohm's law equation with the above equation, two other power equations can be generated. They are

P = I^{2} •
R |
P = V^{2}
/ R |

The standard metric unit of power is the Watt and is dimensionally
equivalent to an Amp•Volt, an Amp^{2}•Ohm, and a
Volt^{2}/Ohm.

**Electricity Costs**

A commerical power company charges households for the energy supplied on a monthly basis. The bill for the services typically states the amount of energy consumed during the month in units of kiloWatt•hours. This unit - a power unit multimplied by a time unit - is a unit of energy. A houlsehold typically pays the bill on the basis of the number of kW•hr of electrical energy consumed during the month. Thus, the task of determining the cost of using a specific appliance for a specified period of time (such as in #18 and #19) is quite straightforward. The power must first be determined and converted to kiloWatts. This power must then be multiplied by the usage time in hours to obtain the energy consumed in units of kW•hr. Finally, this energy amount must be multiplied by the cost of electricity on a $/kW•hr basis in order to determine the cost in dollars.

**Chemistry Connections**

In two of the problems of this set, there is a chemistry connection made. The electrical energy supplied by a circuit is used to heat water (#17) and to vaporize water (#20). Energy is required for each process. The amount of energy is needed to warm water by an amount T is given by the equation

where **m** is the mass of the
water (grams), **C** is the specific
heat capacity of water (4.186 J/g deg) and
**T**
is the temperature change of the water (degrees C).

The amount of energy required to vaporize water is dependent upon the mass of water being vaporized and the heat of vaporization of water. The equation is

where **m** is the mass of water
(kg) and **
H _{vap}** is the heat of vaporization of water (2.26
x 10^6 J/kg).

To be successful on this problem set, you will have to:

- have a comfort with the above equations.
- identify electrical quanities from a careful reading of the problem statement.
- give attention to units and have a plan for converting units in such a manner that the unknown quantity is expressed in the requested units.
- practice the habits of a good problem solver.

**Additional Readings/Study
Aids:**

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.

Resistance and Resistivity | Voltage-Current-Resistance | More Power Equations |

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