# CHARGES IN AN ELECTRIC FIELD

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1 CONTENT POTENTIAL ENERGY AND WORK HSC Physics Module 4 Electricity and Magnetism Electric Field Potential energy, Work and Equipotential Lines CHARGES IN AN ELECTRIC FIELD A charge in an electric field is analogous to an object in a gravitational field An object positioned at higher gravitational potential energy will, if unrestrained, naturally move towards lower potential energy In this situation, work is done by the gravitational force If we move the object from a position of lower gravitational potential energy to higher potential energy, then work is done by the external force used to move it Similarly, a charge in an electric field, will, if unrestrained, naturally move from a higher to lower electric potential energy The reverse movement from a posiition of lower electrical potential energy will require an external force to do the work This is illustrated in the diagram below for a charged particle in an electric field created by two conducting parallel plates The arrows represent the electric field direction The potential difference between two points A and B in the electric field is defined as the amount of work done in moving a unit charge from A to B and is given by Where is the difference in potential energy between point A and B and is the charge of the particle is the potential difference or voltage and is given in units of Volts (V) (This concept is important when we deal with electric circuits) CONTENT EQUIPOTENTIAL LINES The electric potential can be represented graphically by drawing lines that are perpendicular to the electric field These lines represent the points at which the electric potential energy is the same Hence the name equipotential lines The concept of equipotential lines in two dimensions can be extended to equipotental surfaces in three dimensions For two parallel conducting plates, the equipotential surfaces are surfaces parallel to the plates Jeffry Setiadi and Arunima Malik, School of Physics Page 1

2 Electric Field Potential energy, Work and Equipotential Lines Since the electric potential along an equipotential line is the same, no work is done moving a charged particle along the equipotential line However, moving the particle from one equipotential line to another will require work either by the electric field or an external force Below are two more examples of equipotential lines for a single positive charge and for two positive charges Red lines represent the electri field lines while the dashed lines represent the equipotential lines QUESTION 1 Calculate the amount of work done by an electric field (10 V/m) on a particle with a charge of 10 C that moves along an equipotential surface QUESTION 2 - PENCIL Draw the equipotential lines for the positive and negative charges below Jeffry Setiadi and Arunima Malik, School of Physics Page 2

3 Electric Circuits Series and Parallel Circuits WORKED EXAMPLE SERIES CIRCUIT Find the voltage running through each resistor for the series circuit on the right The resistance values are We have three resistors connected in series, and according to Kirchoff s Current Law, the current passing through each resistor is the same Thus, if we know the total current is running through the circuit, we know the current running through each resistor We start by combining the three resistors connected in series into one equivalent resistor The total current running through the whole circuit is: Thus, the voltage across each resistor is: To check if we calculated the voltage correctly, make sure the sum of the voltages across each resistor is equal to (Kirchoff s Voltage Law) WORKED EXAMPLE PARALLEL CIRCUIT Find the current running through each resistor for the parallel circuit on the right The resistance values are The three resistors are connected in parallel and the current entering and exiting the node of the resistors is the same The voltage across each resistor is the same Jeffry Setiadi and Arunima Malik, School of Physics Page 1

4 We combine the three resistors into on equivalent resistor HSC Physics Module 4 Electricity and Magnetism The total current running through the whole circuit is: The voltage running through each resistor is and thus the current through each resistor is: Check that the sum of the currents is equal to the total current: Observe that adding resistances (eg, 2 and ( ) in parallel decreases the total resistance WORKED EXAMPLE COMBINED SERIES AND PARALLEL CIRCUIT Find the voltage and current running through each resistor for the series-parallel circuit on the right The resistance values are For this circuit, the current flows through in series and then splits on a node to and in parallel Much like solving the series and parallel only circuit we want to reduce the circuit into a one resistor circuit We first combine the resistors connected in parallel and then combine the resistors in series Pictorially we perform the transformation below Jeffry Setiadi and Arunima Malik, School of Physics Page 2

5 Below is the working out of combining the resistors as shown above The total current is: Once we have the total current, we work our way back and expand the circuit to get the current and voltages The current is constant for resistors connected in series Thus flows through both and The voltage across and is: Check sum of voltage is equal to : Now that we have the voltage running through we can calculate the current through and Check that the sum of the currents is equal to Summary of the current and voltages: Voltage (V) Current (A) Jeffry Setiadi and Arunima Malik, School of Physics Page 3

6 CONTENT OHM S LAW HSC Physics Module 4 Electricity and Magnetism Electric Circuits Series and Parallel Circuits ELECTRIC CIRCUITS Ohm s law relates the voltage (ie potential difference V) with the current and resistance of an ideal conductor The law states that where is the currrent and is the resistance of the conductor The units for current and resistance is Ampere ( ) and Ohm ( ) respectively The current is a measure of the flow of electrons in the circuit and is inversely proportional to the resistance Hence, a large resistance will reduce the flow of electrons As an analogy, this is similar to water flowing through a tube A narrow tube will reduce the amount of water flowing through whilst larger tube will allow more water to flow through As a simple example, consider a one resistor circuit connected to a battery on the right The battery provides a source of 9 V, and the resistance of the resistor is 2 With the information we have, we can calculate the current flowing through the resistor using Ohm s law The current flowing through the 2 resistor is If the resistance is doubled to the current is reduced to 225 It is important to note that resistors can either be ohmic or nonohmic Ohmic resistors have a constant resistance, hence in the equation above, if a resistor is an ohmic resistor with a constant resistance, then the graph of velocity against time is a straight line In the case of non-ohmic resistors, where resistance in a resistor varies with, say current, then the graph of velocity against time is not a straight line In the case of latter, the resistor is said not to obey Ohm s Law CONTENT SERIES CIRCUITS For a single resistor circuit like the circuit above applying Ohm s law is trivial However, when more than one resistor is connected in either series or parallel, then Ohm s law cannot be applied the same way For resistors connected in series (ie connected next to each other) the current flowing through each resistor is the same, but the voltage is divided amongst the resistors For the circuit below with an incoming voltage of, the voltage drop after passing through is Then the voltage drop after passing through is and similarly for Thus, after passing through each resistor the potential of the current flow decreases If all resistors in series have same resistance, then the voltage drop across each resistor is the same However, in the case of resistors with different resistance, the resistor with a greater resistance will result in the greatest voltage drop We can also use the analogy of water flowing from a higher to lower potential energy Each drop in potential energy represents the electric current flowing through a resistor When the water flows horizontally, there is no change in potential energy (like an electric current flowing through the wire) The sum of each potential energy drop is equal to the total potential energy the water is at initially In electric circuits, this is called the Kirchoff s Jeffry Setiadi and Arunima Malik, School of Physics Page 1

7 Voltage Law (KVL), which states that the total potential difference in a closed circuit is equal to zero (ie conservation of energy) The total resistance of resistors connected in series can be calculated by summing up the resistance value of each resistor The total resistance is also called the equivalent resistance, and we can reduce the multi-resistor circuit to one equivalent resistor CONTENT PARALLEL CIRCUITS For resistors connected in parallel the voltage flowing through each resistor is the same but the current is divided, which is the opposite behaviour to resistors connected in series Consider the image of water tubes below where the flow of water splits into three tubes of different widths (point A) The amount of water flowing through tube 1 to 3 depends on the width of the tubes As the water reaches the exit junction, point B, the current flow returns to the same state as the current it enters in at point A The potential difference, however, is the same in tubes 1 to 3 Similarly, for an electric circuit, the sum of the electric current flowing through each resistor connected in parallel is equal to the total current flowing into or out of the node/junction This behaviour is known as the Kirchoff s Current Law (KCL), which states that the total current entering a node is equal to the total current exiting the node (ie conservation of charge) Mathematically this is written as The equivalent resistance of resistors connected in parallel is calculated by summing up the reciprocal of each resistance value Transforming the resistors connected in parallel into one equivalent resistor SUMMARY The table below summarises the general behaviour of current and voltage in circuits Jeffry Setiadi and Arunima Malik, School of Physics Page 2

8 Resistor Connection Voltage Current Series Divided Constant Parallel Constant Divided HSC Physics Module 4 Electricity and Magnetism WORKED EXAMPLE SERIES CIRCUIT Find the voltage running through each resistor for the series circuit on the right The resistance values are We have three resistors connected in series, and according to Kirchoff s Current Law, the current passing through each resistor is the same Thus, if we know the total current is running through the circuit, we know the current running through each resistor We start by combining the three resistors connected in series into one equivalent resistor The total current running through the whole circuit is: Thus, the voltage across each resistor is: To check if we calculated the voltage correctly, make sure the sum of the voltages across each resistor is equal to (Kirchoff s Voltage Law) WORKED EXAMPLE PARALLEL CIRCUIT Find the current running through each resistor for the parallel circuit on the right The resistance values are Jeffry Setiadi and Arunima Malik, School of Physics Page 3

9 The three resistors are connected in parallel and the current entering and exiting the node of the resistors is the same The voltage across each resistor is the same We combine the three resistors into on equivalent resistor The total current running through the whole circuit is: The voltage running through each resistor is and thus the current through each resistor is: Check that the sum of the currents is equal to the total current: Observe that adding resistances (eg, 2 and ( ) in parallel decreases the total resistance WORKED EXAMPLE COMBINED SERIES AND PARALLEL CIRCUIT Find the voltage and current running through each resistor for the series-parallel circuit on the right The resistance values are For this circuit, the current flows through in series and then splits on a node to and in parallel Much like solving the series and parallel only circuit we want to reduce the circuit into a one resistor circuit We first combine the resistors connected in parallel and then combine the resistors in series Pictorially we perform the transformation below Jeffry Setiadi and Arunima Malik, School of Physics Page 4

10 Below is the working out of combining the resistors as shown above The total current is: Once we have the total current, we work our way back and expand the circuit to get the current and voltages The current is constant for resistors connected in series Thus flows through both and The voltage across and is: Check sum of voltage is equal to : Now that we have the voltage running through we can calculate the current through and Check that the sum of the currents is equal to Jeffry Setiadi and Arunima Malik, School of Physics Page 5

11 Summary of the current and voltages: Voltage (V) Current (A) Jeffry Setiadi and Arunima Malik, School of Physics Page 6

12 Electric Circuits Heating Effects HEATING EFFECTS OF ELECTRIC CURRENTS CONTENT CONSERVATION OF ENERGY An electric current that passes through an electric component will have its potential energy dropped by a certain amount (potential difference or voltage) Following the circuit on the right, the electric current flows from a source starting from the positive terminal As the current passes through the resistor, the potential energy of the electric current drops between point A and B Finally, the current flows to the negative terminal completing the circuit Following the law of conservation of energy, the drop in potential energy from point A to B must be transformed into a different kind of energy (the energy loss across the resistor cannot be destroyed) For a resistor, the electrical energy is transferred into heat (electrical to thermal) Other devices connected similarly will convert the electrical energy differently Examples include electric current passing through a motor connected to a mechanical load, light bulb and a storage battery The electrical energy is transferred as work done on the mechanical load (electrical to mechanical), converted to light (electrical to electromagnetic) and transferred to a stored chemical in the battery (electrical to chemical) CONTENT POWER IN CIRCUITS The transfer of energy is quantitatively described with a variable called power, The units for is joule/sec or Watts, hence it is a measure of the amount of energy transferred per second or unit time The power transferred from electric current is given by the voltage multiplied by the current To check if the expression above is correct, we will look at the units of both voltage and current Voltage is defined as the amount of work done per unit charge (joule/coulomb), and current is the flow of charge per unit time (coulomb/sec) Thus, if we have 1 Volt and 1 Amp the multiplication of the two is (1) Jeffry Setiadi and Arunima Malik, School of Physics Page 1

13 Electric Field Heating Effects which is the unit for power We can combine Ohm s law with the power equation above to get the dissipation rate of electrical energy across a resistor Ohm s law is usually written as and if we substitute Ohm s law into equation (1) we get If instead, we write Ohm s law as then power dissipated across the resistor is ( ) (3) In the case of a series circuit, where current is the same across all resistors, the resistor with the greater resistance will have a greater power dissipation In other words, if current stays the same in equation 2), then the greater the resistance, the greater the power dissipation In the case of a parallel circuit, however, where voltage across the resistors stays the same, the resistor with the greater resistance will have a smaller power dissipation This can be explained by visualising voltage as a constant in equation 3) and varying the resistance It is important to remember that equation (1) is the rate of transfer of electrical energy to all forms of energy Equation (2) and (3) gives the rate of transfer of electrical energy to thermal energy for an electrical component with resistance This behaviour of electric current in circuits is used in the design of electric heaters (2) QUESTION 1 A 30 Ω resistor is connected to a 90 V battery as shown on the circuit on the right a) How much work is done on moving an electron across the resistor? b) What is the rate of energy transferred into heat across the resistor? c) How much thermal energy is produced in 1 hour? ans: (a) 90 J (b) 27 W (c) 97 kj QUESTION 2 Heat is generated in a resistor at a rate of 50 W If the current running through the resistor is 15 A what is the resistance value of the resistor? What is potential difference across the resistor? ans: R=222 Ω and V=333 V QUESTION 3 The standard voltage in Australia is 230 V and if a 1000 W heater is plugged into an outlet, find: a) The amount of current flowing through the heater b) The resistance of the heater c) The total cost of continuously running the bulb for seven days if it cost 125 cents/(kwh) ans: (a) 43 A (b) 535 Ω (c) \$2100 QUESTION 4 INTERNET RESEARCH a) Investigate the physics of fireworks in terms of energy conversions How many energy conversions can you list? b) Research the power consumption of common household appliances Pick one appliance and study the energy rating of that appliance What does the star rating mean? Jeffry Setiadi and Arunima Malik, School of Physics Page 2

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