ChemPhys 173/273

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

I = Q / t

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.

Qelectron = 1.6 x 10-19 C

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 m2. 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.

A = • R2 = • D2 / 4

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

V = I • R .

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

P = E / t

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

P = I • V

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

 P = I2 • R P = V2 / R

The standard metric unit of power is the Watt and is dimensionally equivalent to an Amp•Volt, an Amp2•Ohm, and a Volt2/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

E = m • C • T

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

E = m • Hvap

where m is the mass of water (kg) and Hvap 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.

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.

Electric Potential Difference | Electric Current | Electric Power |

View Sample Problem Set.

 Problem Description Audio Link 1 Determination of the amount of charge passing a point on a circuit over a stated period of time. 2 Determination of the amount of charge passing through a bulb on a circuit over a stated period of time if given the amount passing through another bulb in the same period of time. 3 Determination of the current if given the number of electrons passing a point in a given amount of time. 4 Determination of the current if given the number of electrons passing a point in a given amount of time. 5 Determination of the voltage impressed between two points of known resistance in order to allow a stated current value to be present. 6 Determination of the voltage impressed between two points of known resistance in order to allow a stated current value to be present. 7 Determination of the amount of charge passing a point on a circuit over a stated period of time. 8 Determinatio of the resistance of a television if given its current and the voltage drop across it. 9 Use of Ohms law equation to relate current, resistance and voltage. 10 Determination of the length of a wire if given the resistance, material and cross-sectional area. 11 Determination of the length of a wire if given the resistance, material and diameter. 12 Determination of the length of a gold wire which would provide the same resistance as an iron wire of the same radius. 13 Determination of the resistance of a toaster from its power rating at 120 Volts. 14 Determination of the current supplied to a generator if given its power consumption rate and its voltage output. 15 Determination of the power of a CD player if given the current and voltage supplied to it. 16 Determination of the cross-sectional area of the Tungsten heating element on a heater if given its length, power output and the voltage supplied to it. 17 Use of Q = m•C•T to determine the resistance of an immersion heater from its voltage source and the rate at which it increases the temperature of known quantity of water. 18 Determination of the cost of a watching a world series if given the time, the power and the electricity cost. 19 Determinatio of the cost of operating a hair dryer for a specific amount of time if given the power and the electricity cost. 20 Use of heat of vaporization value to determine the resistance of a vaporizer which causes a known volume of water to vaporize in a stated period of time when operating at a known voltage.

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