EGR 1980 Recitation Worksheet 2 Using Scientific Notation in Engineering Mathematics H Griffith 17 January 2014 Introduction In class we reviewed the topic of exponents along with the associated rules utilized for simplifying exponential expressions including 1 the product rule 2 the quotient rule and 3 the power rule One of the many applications of these rules is in performing arithmetic for both the very large and small values oftentimes encountered in engineering which are expressed using scientific notation When a number is written in scientific notation it contains two unique parts 1 the coefficient and 2 the order of magnitude or exponent By convention the coefficient of a number written in scientific notation should be greater than 1 but less than 10 An example of a number written in scientific notation is shown below Very small numbers are represented in scientific notation with negative orders of magnitude possessing large absolute values In contrast very large numbers are represented in scientific notation with large positive numbers Another important function of representing physical values in scientific notation is ease of integration with the international system of units SI Particularly scientists and engineers can represent quantities whose magnitude varies substantially using a common base unit simply adjusted with an appropriate order of magnitude By convention orders of magnitude are replaced by corresponding prefixes The prefixes which you should remember for this class along with their corresponding orders of magnitude are tabulated below Prefix Giga Mega Kilo milli micro nano Abbreviation G M K m Pronounced mu n Order of Magnitude Today we ll demonstrate how to appropriately utilize these prefixes and exponential rules using basic circuit analysis computations Circuit Analysis Ohm s Law and the Power Rule Last recitation we defined the concept of resistance and developed techniques for describing the equivalent resistance of electrical networks consisting of more than one electrical load Today we ll take a closer look at electrical circuits which consist of both electrical loads and sources In order to do this we must look at the two values which influence the rate at which an electrical source delivers energy to a circuit which is referred to as the supplied power Recall from lecture that power is measured in units of Watts which is abbreviated by the capital letter W By definition 1 Watt is equivalent to delivering 1 Joule of energy per second Mathematically this is written as To begin let s consider circuits whose source of input energy is referred to as an ideal voltage source By definition an ideal voltage source is capable of maintaining the rated terminal voltage across its nodes regardless of the type of load to which it is supplying power to As was discussed in lecture voltage describes the amount of work which would be necessary in order to move a bundle of positive electrical charge referred to as a Coulomb from the negative point of reference of the voltage measurement ie the negative terminal to the positive point of reference of the voltage measurement ie the positive terminal Note that work is required to accomplish this task as positive charge is naturally repelled from areas of high potential Schematically an ideal voltage source is represented as follows As was the case with resistors the appropriate schematic of an ideal voltage source consists of three parts 1 the reference designator 2 the rated voltage value and 3 the schematic symbol Note that voltage is measured in units of Volts which is abbreviated by the capital letter V By definition 1 Joule unit of energy is required to move 1 Coulomb of charge bundle of charge abbreviated by the capital letter C across a 1 Volt potential difference or gradient Mathematically this is expressed as follows You try Using the technique of proportions described in class determine the amount of charge which may be moved across a 10 Volt potential difference using only 1 Joule of energy When connected across a resistive load a complete circuit is formed thereby establishing the flow of electrical charge which is referred to as electrical current Electrical current is measured in units of Amperes abbreviated by the capital letter A By definition 1 Ampere of current represents the flow of 1 Coloumb of charge through an entire cross section of a conductor in 1 second This is represented mathematically as The amount of electrical current produced in a resistive network is proportional to the rated voltage of the source This is property is described by the physical principle known as Ohm s Law This relationship can be expressed algebraically as is shown below Recall that using algebra the above expression may be rearranged in various forms depending upon which quantities are known in a particular circuit For example Ohm s Law solved for current as a function of voltage and resistance as given by Note that using Ohm s Law we can develop relationships between the three basic circuit quantities of voltage current and resistance Namely You try An engineer is attempting to describe the loading characteristics of a small heating element In order to do this he connects a D cell battery terminal voltage rating of 1 5 Volts to the load and forms a complete circuit Upon taking measurements he measures the voltage across the load as 1 49 Volts and the associated current as 1 Ampere 1 Draw and electrical circuit representative of the above scenario 2 Manipulate Ohm s Law in a form suitable for describing the equivalent resistance of the load 3 Solve for the equivalent resistance of the load In order to describe the power or rate at which an electrical source provides energy to a load we must consider both the terminal voltage measured across the source as well as the electrical current leaving the source in the presence of the load Physically we can show that the power provided by a source is calculated as the product of these two quantities as described by the equation below As was the case for Ohm s Law the above relationship may be manipulated using algebra in order to solve for any one of the three quantities described in the equation Through substitution Ohm s Law and the power equation may be combined in order to yield formulas for power which are given in terms of solely the source voltage or current and corresponding load resistance This is useful in practice as the two types of measurements voltage and current