Chapter 6 Current, esistance, and Direct Current Circuits Electric Current Whenever electric charges of like signs move, an electric current is said to exist The current is the rate at which the charge flows through this surface Look at the charges flowing perpendicularly to a surface of area A Q I t The SI unit of current is Ampere (A) A = C/s Electric Current, cont The direction of the current is the direction positive charge would flow This is known as conventional current direction In a common conductor, such as copper, the current is due to the motion of the negatively charged electrons It is common to refer to a moving charge as a mobile charge carrier A charge carrier can be positive or negative Was decided before it was realized that electrons are negatively charged Current and Drift Speed Charged particles move through a conductor of cross-sectional area A n is the number of charge carriers per unit volume n A Δx is the total number of charge carriers Current and Drift Speed, cont The total charge is the number of carriers times the charge per carrier, q ΔQ total = (n (volume)) q carrier = (n A cross sectional Δx) q The drift speed, v d, is the speed at which the carriers move v d = Δx/ Δt ewritten: ΔQ = (n A v d Δt) q Finally, current, I = ΔQ/Δt = nq c v d A cs Current and Drift Speed, final If the conductor is isolated, the electrons undergo random motion When an electric field is set up in the conductor, it creates an electric force on the electrons and hence a current
Charge Carrier Motion in a Conductor The zig-zag black line represents the motion of charge carrier in a conductor The net drift speed is small The sharp changes in direction are due to collisions The net motion of electrons is opposite the direction of the electric field Electrons in a Circuit The drift speed is much smaller than the average speed between collisions When a circuit is completed, the electric field (NOT THE ELECTON) travels with a speed close to the speed of light Although the drift speed is on the order of 0-4 m/s the effect of the electric field is felt on the order of 0 8 m/s 2- Electric Current A battery uses chemical reactions to produce a potential difference between its terminals. It causes current to flow through the flashlight bulb similar to the way the person lifting the water causes the water to flow through the paddle wheel. 2- Electric Current A battery that is disconnected from any circuit has an electric potential difference between its terminals that is called the electromotive force or emf: emember despite its name, the emf is an electric potential, not a force. The amount of work it takes to move a charge ΔQ from one terminal to the other is: esistance In a conductor, the voltage applied across the ends of the conductor is proportional to the current through the conductor The constant of proportionality is the resistance of the conductor esistance, cont Units of resistance are ohms (Ω) Ω = V / A esistance in a circuit arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor V I 2
Georg Simon Ohm 787 854 Formulated the concept of resistance Discovered the proportionality between current and voltages Ohm s Law Experiments show that for many materials, including most metals, the resistance remains constant over a wide range of applied voltages or currents This statement has become known as Ohm s Law ΔV = I Ohm s Law is an empirical relationship that is valid only for certain materials Materials that obey Ohm s Law are said to be ohmic Ohm s Law, cont An ohmic device The resistance is constant over a wide range of voltages The relationship between current and voltage is linear The slope is related to the resistance Ohm s Law, final Non-ohmic materials are those whose resistance changes with voltage or current The current-voltage relationship is nonlinear A diode is a common example of a non-ohmic device esistivity The resistance of an ohmic conductor is proportional to its length, L, and inversely proportional to its cross-sectional area, A L A ρ is the constant of proportionality and is called the resistivity of the material 2-2 esistance and Ohm s Law The difference between insulators, semiconductors, and conductors can be clearly seen in their resistivities: 3
Temperature Variation of esistivity For most metals, resistivity increases with increasing temperature With a higher temperature, the metal s constituent atoms vibrate with increasing amplitude The electrons find it more difficult to pass through the atoms Temperature Variation of esistance Since the resistance of a conductor with uniform cross sectional area is proportional to the resistivity, you can find the effect of temperature on resistance o [ (T T o )] Temperature Variation of esistivity, cont For most metals, resistivity increases approximately linearly with temperature over a limited temperature range o [ (T To )] ρ is the resistivity at some temperature T ρ o is the resistivity at some reference temperature T o T o is usually taken to be 20 C is the temperature coefficient of resistivity 2-2 esistance and Ohm s Law In general, the resistance of materials goes up as the temperature goes up, due to thermal effects. This property can be used in thermometers. esistivity decreases as the temperature decreases, but there is a certain class of materials called superconductors in which the resistivity drops suddenly to zero at a finite temperature, called the critical temperature T C. Electrical Energy and Power Superconductors will be discussed in AP Physics 2 In a circuit, as a charge moves through the battery, the electrical potential energy of the system is increased by ΔQΔV The chemical potential energy of the battery decreases by the same amount As the charge moves through a resistor, it loses this potential energy during collisions with atoms in the resistor The temperature of the resistor will increase 4
2-3 Energy and Power in Electric Circuits When a charge moves across a potential difference, its potential energy changes: Therefore, the power it takes to do this is Energy Transfer in the Circuit Consider the circuit shown Imagine a quantity of positive charge, Q, moving around the circuit from point A back to point A Energy Transfer in the Circuit, cont Point A is the reference point It is grounded and its potential is taken to be zero As the charge moves through the battery from A to B, the potential energy of the system increases by Q V The chemical energy of the battery decreases by the same amount Energy Transfer in the Circuit, final As the charge moves through the resistor, from C to D, it loses energy in collisions with the atoms of the resistor The energy is transferred to internal energy When the charge returns to A, the net result is that some chemical energy of the battery has been delivered to the resistor and caused its temperature to rise Electrical Energy and Power, cont The rate at which the energy is lost is the power Q V I V t From Ohm s Law, alternate forms of power are 2-3 Energy and Power in Electric Circuits When the electric company sends you a bill, your usage is quoted in kilowatt-hours (kwh). They are charging you for energy use, and kwh are a measure of energy. 2 V I 2 5
Electrical Energy and Power, final The SI unit of power is Watt (W) I must be in Amperes, in ohms and V in Volts The unit of energy used by electric companies is the kilowatt-hour This is defined in terms of the unit of power and the amount of time it is supplied kwh = 3.60 x 0 6 J Electrical Activity in the Heart Will Be Discussed in AP Physics 2 Meters in a Circuit Ammeter 2-8 Ammeters An ammeter is a device for measuring current, and a voltmeter measures voltages. The current in the circuit must flow through the ammeter; therefore the ammeter should have as low a resistance as possible, for the least disturbance. An ammeter is used to measure current In line with the bulb, all the charge passing through the bulb also must pass through the meter Meters in a Circuit Voltmeter 2-8 Voltmeters A voltmeter measures the potential drop between two points in a circuit. It therefore is connected in parallel; in order to minimize the effect on the circuit, it should have as large a resistance as possible. A voltmeter is used to measure voltage (potential difference) Connects to the two ends of the bulb 6
More Detailed EMF Will be Discussed in AP Physics 2 esistors in Series When two or more resistors are connected end-to-end, they are said to be in series They can be replaced by a single equivalent resistance without changing the current in the circuit. The current is the same in all resistors because any charge that flows through one resistor flows through the other The sum of the potential differences across the resistors is equal to the total potential difference across the combination esistors in Series, cont Potentials add ΔV = I + I 2 = I ( + 2 ) Consequence of Conservation of Energy The equivalent resistance has the effect on the circuit as the original combination of resistors Equivalent esistance Series eq = + 2 + 3 + The equivalent resistance of a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any of the individual resistors Equivalent esistance Series: An Example Four resistors are replaced with their equivalent resistance esistors in Parallel The potential difference across each resistor is the same because each is connected directly across the battery terminals esistors are in parallel when they are across the same potential difference; they can again be replaced by a single equivalent resistance: The current, I, that enters a point must be equal to the total current leaving that point I = I + I 2 The currents are generally not the same Consequence of Conservation of Charge 7
Equivalent esistance Parallel, Example Equivalent resistance replaces the two original resistances Household circuits are wired so the electrical devices are connected in parallel Circuit breakers may be used in series with other circuit elements for safety purposes Equivalent esistance Parallel Equivalent esistance eq 2 3 The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistance The equivalent is always less than the smallest resistor in the group ules to emember Combine all resistors in series They carry the same current The potential differences across them are not the same The resistors add directly to give the equivalent resistance of the series combination: eq = + 2 + ules to emember Combine all resistors in parallel The potential differences across them are the same The currents through them are not the same The equivalent resistance of a parallel combination is found through reciprocal addition: eq 2 3 2-4 esistors in Series and Parallel If a circuit is more complex, start with combinations of resistors that are either purely in series or in parallel. eplace these with their equivalent resistances; as you go on you will be able to replace more and more of them. Equivalent esistance Complex Circuit 8
Gustav Kirchhoff 824 887 Invented spectroscopy with obert Bunsen Formulated rules about radiation 2-5 Kirchhoff s ules More complex circuits cannot be broken down into series and parallel pieces. For these circuits, Kirchhoff s rules are useful. The junction rule is a consequence of charge conservation; the loop rule is a consequence of energy conservation. Statement of Kirchhoff s ules Junction ule The sum of the currents entering any junction must equal the sum of the currents leaving that junction A statement of Conservation of Charge Loop ule The sum of the potential differences across all the elements around any closed circuit loop must be zero A statement of Conservation of Energy More About the Junction ule I = I 2 + I 3 From Conservation of Charge Diagram b shows a mechanical analog Setting Up Kirchhoff s ules Assign symbols and directions to the currents in all branches of the circuit If a direction is chosen incorrectly, the resulting answer will be negative, but the magnitude will be correct When applying the loop rule, choose a direction for transversing the loop ecord voltage drops and rises as they occur More About the Loop ule Traveling around the loop from a to b In a, the resistor is transversed in the direction of the current, the potential across the resistor is I In b, the resistor is transversed in the direction opposite of the current, the potential across the resistor is +I 9
Loop ule, final In c, the source of emf is transversed in the direction of the emf (from to +), the change in the electric potential is +ε In d, the source of emf is transversed in the direction opposite of the emf (from + to -), the change in the electric potential is -ε Junction Equations from Kirchhoff s ules Use the junction rule as often as needed, so long as, each time you write an equation, you include in it a current that has not been used in a previous junction rule equation In general, the number of times the junction rule can be used is one fewer than the number of junction points in the circuit Loop Equations from Kirchhoff s ules The loop rule can be used as often as needed so long as a new circuit element (resistor or battery) or a new current appears in each new equation You need as many independent equations as you have unknowns Circuits Containing Capacitors Will Be Discussed in AP Physics 2 0