# Notes on Electricity (Circuits)

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1 A circuit is defined to be a collection of energy-givers (active elements) and energy-takers (passive elements) that form a closed path (or complete path) through which electrical current can flow. The active elements are voltage sources (like batteries) that provide energy, and the passive elements are resistors (or light bulbs, ipods, computers, etc.) that use the energy. Current As you remember from previous studies of Chemistry, all matter is made of atoms. All atoms are made of protons, neutrons, and electrons. The protons have a positive charge; the electrons have a negative charge; the neutrons have no charge. The charge of protons and electrons are often referred to as + and -, respectively. This is done to indicate that they have equal and opposite charges. The actual units for electrical charge are called Coulombs. One Coulomb represents a very large charge. As a matter of fact, it takes 6,250,000,000,000,000,000 electrons to make one Coulomb of charge. The movement of electrical charges is called electrical current. Current can be thought of as how fast charges flow or how many Coulombs of charge pass a point every second. The units for current are Amperes (or Amps, for short). One Ampere of current is equal to one Coulomb/second ( A = C/s). The symbol used for current in an equation is I. By definition, current flows in the direction that positive charges would move. It should be noted that a current of 70 ma can be fatal, so it is very important that you pay close attention to the prefixes that go with each unit. The difference between 0 ma and 0 A is NOT a small one! Voltage Voltage is a measure of the difference in electrical potential energy (per unit of charge) between two points in a circuit. Voltage is often called the potential difference. The units for voltage are Volts (V). Since the units for energy are Joules and the units for charge are Coulombs, one Volt is equal to one Joule/Coulomb ( V = J/C). The symbol used for voltage in an equation is V. (For the sake of developing an initial understanding of circuits, we will think of voltage simply as being the difference in energy between two points.) All batteries have a positive terminal and a negative terminal. By definition, the positive terminal has more energy. For a 2-Volt battery, the difference in energy between the positive and the negative terminals is 2 Volts, with the positive terminal having more energy. The voltage for a D-cell battery is.5 Volts, which means that the positive terminal has.5 Joules more energy (per Coulomb of charge) than the negative terminal. Since voltage is the difference in energy between two points, we never talk about the voltage of one point in a circuit. We always refer to the voltage between two points.

2 Resistance Resistance is a measure of how difficult it is for electrical current to pass through something, or how much something resists the flow of current. Electrical conductors tend to have a very low resistance, while electrical insulators have a very high resistance. Resistance is measured in units of Ohms (Ω). Resistors are energy-takers. When electrical current passes through a resistor, some electrical energy is lost and converted into other forms of energy such as heat, light, or sound. Light bulbs, radios, TVs, and other appliances can be treated as if they were resistors, since what they do is to convert electrical energy into some other form of energy. Ohm s Law Ohm s Law relates voltage, current, and resistance in a simple equation. Ohm s Law states that the voltage between two points in equal to the current flowing between the points multiplied by the resistance between the points. V = I x R Looking at the units for the values in Ohm s Law shows that (Volts) = (Amps) x (Ohms). If the equation is solved for resistance, the units show that (Ohms) = (Volts)/(Amps). Note that it is common for current to be measured in milliamps (ma) and resistance in kiloohms (kω). Power Power is defined to be the rate at which (how fast) energy is generated or the rate at which (how fast) energy is used. One way to express power is Energy Power time The units for energy are Joules, and the units for time are seconds. So, power has units of Joules/second, which are called Watts (W). In a circuit, the power being used or generated by a particular element can be found by multiplying the current and voltage for that element. P = I x V If we look at the units for this equation, (Amps) x (Volts) equal (Watts). ( Watts ) ( Amps) x( Volts ) ( C / s) x( J / C) ( J / s)

3 Series & Parallel The elements that make up a circuit can be connected in different configurations. The regions of the circuit where three or more elements are connected are called nodes. Since a node connects more than two elements, when current enters the node it will split, and part of it will go to each of the elements on the other side of the node. Every point within a given node has the same amount of electric potential energy. Different nodes have different amounts of electric potential energy. The difference in the energy (per charge) between two nodes is the voltage. Elements are connected in series when there is no node in between them. Since there is no node between them (no place for the current to split), elements that are in series with each other must have the same current. Elements are connected in parallel when they are connected between the same two nodes. Since the elements connect the same two nodes and since difference in energy between the two nodes is the voltage, elements in parallel must have the same voltage. Ex: The circuit shown to the right has two nodes, as shown by the red lines. The two 4 kω resistors both connect Node and, therefore, they are connected in parallel to one another and would have the same voltage. The 8 Volt source and the 2 kω resistor are connected to one another with no node in between them. So, the battery and the 2 kω resistor are in series with one another and would have the same current. Node When every element in a circuit is connected in series (as shown to the left), there is only one path through which current can flow. If one of the elements is removed or if it breaks, then the circuit will be broken and no current will flow to any of the other elements. When every element in a circuit is connected in parallel (as shown to the right), there are multiple paths through which current can flow. If one of the elements is removed or if it breaks, then there are still other paths through which current can flow. In this case, the other elements will continue to function since current will still be flowing. Node Any element or group of elements that connects one node to another node is called a branch. If a branch consists of more than one element, then those elements are in series with each other. While it is possible for individual elements to be in parallel with each other, it is also possible for branches to be in parallel with other branches or for individual elements to be in parallel with branches.

4 3Ω Node 4Ω In the circuit to the left, there are two nodes that are highlighted in red. 2 V There are three branches between the two nodes. The branch on the 2Ω left consists of a single 3 Ω. The branch in the middle consists of the 2 V source and a 4 Ω resistor that are in series. And, the branch on 4Ω the right consists of a 4 Ω resistor in series with a 2 Ω resistor. None of the individual elements are in parallel with each other, because no two of the elements are directly connected between the two nodes. (The 3 Ω resistor is the only element that directly connects the two nodes.) However, all three of the branches are in parallel with each other, since all three branches connect the top node to the bottom node. Kirchhoff s Laws Nodes are simply wires that connect circuit elements, it is impossible for current to be stored or generated within the wire itself. Given that, the total current that enters a node must equal the total current that leaves the node. This fact is known as Kirchhoff s Current Law. Applying Kirchhoff s Current Law to the circuit below yields the equation I = I 2 + I 3 8Ω Node I I I 3 4Ω V I 2 V 3 2 V V 2 6Ω V 4 8Ω Similarly, Kirchhoff s Voltage Law states that the total voltage (difference in potential) around a complete loop (starting and finishing at the same point) must equal zero. Since the two points being compared are the starting point and the finishing point, the difference in potential must be zero since the two points are the same point. There are three complete loops in the circuit above that start and finish at. Applying Kirchhoff s Voltage Law to each of these circuits yields the following three equations: 2 V + V + V 2 = 0 OR 2 V = V + V 2 () 2 V + V + V 3 + ( V 4 ) = 0 OR 2 V = V + V 3 + V 4 (2) V 2 + V 3 + ( V 4 ) = 0 OR V 2 = V 3 + V 4 (3)

5 Equivalent Resistance Resistors in Series: It is possible to replace resistors connected in series with one equivalent resistor. The value of the equivalent resistor is equal to the sum of the resistors in series R eq = ΣR i. The idea is that the group of resistors in series could be removed, and the equivalent resistance could be placed where the group of resistors was. The derivation of this is given in the following example. In the simple series circuit to the right, the voltage source, V s, will produce a current, I s. That current will then flow through all three resistors, since they are in series with each other. As the current flows through each resistor, there will be a loss of energy resulting in a potential difference (or voltage) across each resistor. This voltage can be calculated using Ohm s Law, as shown below. Vs R R2 V = I s R V 2 = I s R 2 V 3 = I s R 3 R3 Applying Kirchhoff s Voltage Law to this circuit yields the following: Resistors in Parallel: V s = V + V 2 + V 3 V s = I s R + I s R 2 + I s R 3 V s = I s (R + R 2 + R 3) V s = I s R eq It is possible to replace resistors connected in parallel with one equivalent resistor. The example that follows provides the derivation for the equivalent resistance of a group of parallel resistors. In the circuit shown, all four elements are in parallel between the two nodes. The voltage between the nodes is V s. The current produced by the source, I s, will split when it enters the top node, with some of the current going through each of the three resistors. Applying Kirchhoff s Current Law to the top node yields the following: Vs R R2 R3 I s = I + I 2 + I 3 I s = V s R + V s R 2 + V s R 3 I s = V s R + R 2 + R 3

6 Rearranging the last equation gives an equation in the form of Ohm s Law. V s = I s R + R + 2 R 3 The term in brackets represents the equivalent resistance of the three resistors in parallel. The equation below gives another way to show this relationship. R eq = R + R 2 + R 3 + = R i where R, R 2, R 3, and R i represent the individual resistors that are in parallel. Since the individual resistors are connected between the same two nodes, the equivalent resistor is placed between those same two nodes. Note: If there are only two resistors, then the equivalent resistance is equal to the product of the resistors in parallel divided by their sum. R eq = R R 2 R + R 2 It is very common for circuits to have combinations of resistors in series and parallel. These equivalent resistance concepts can be applied to subsets of the overall circuit to simplify the circuit incrementally. Each simplified version of the circuit can then be simplified further, until a single equivalent resistance is obtained. (See the example on the website for how this is done.)

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