AP Physics C - E & M

AP Physics C - E & M Current and Circuits 2017-07-12 www.njctl.org

Electric Current Resistance and Resistivity Electromotive Force (EMF) Energy and Power Resistors in Series and in Parallel Kirchoff's Rules Circuit Measuring Devices RC Circuits Table of Contents: Current and Circuits Click on the topic to go to that section. Move any photo or image in this presentation to reveal a link to its source, providing attribution and additional information.

Electric Current Return to Table of Contents

Electric Current Electric Current is defined as the movement of charge from one region of space to another, and is denoted by a capital I. In conductors, electrons move around freely, not attached to any specific atom, but since they are going off in all different directions, the net current in any normal conductor such as copper would be zero. However if an electric field were passed through the copper wire, the free charges would begin to move in one specific direction resulting in a current.

Electric Current The net motion of these charges due to the force exerted by the Electric Field is called drift velocity, v d. Even though individual speeds of the electrons are on the order of 10 6 m/s, the random nature of their movement and their collisions with the other electrons and ions in the conductor, slow their drift velocity to 10-3 m/s. The Electric Field propagates at nearly the speed of light through the material, but the physical movement of the electrons is so much slower.

Electric Current + + + _ + + Before Electric Field _ + _ + + + _ + After Electric Field Once the Electric Field is turned on, there is a net force to the right on the positively charged particles, and a net force to the left on the negative particles. The negative charges (electrons) would actually move, and the positive charges (nuclei) would stay put.

Electric Current + + + _ + + Before Electric Field _ + _ + + + _ + After Electric Field Circuit Analysis defines current as the direction that positive charges move - conventional current. This is a legacy from Ben Franklin who assigned the negative quality to the charge left on the glass after being rubbed by silk. It should have been positive. Click here for an interesting article on this naming convention.

Electric Current Both positive and negative charges can generate an electric current, but only when they pass through certain materials. In metals only electrons can flow, but in ionized gases, such as plasma, both electrons and positive ions are able to flow.

Electric Current + + + + In the diagram above, positive charges are flowing through a specific area in a certain amount of time. The rate of flow of charges is defined as current and is represented as: The unit for current is coulombs/second, or ampere, named after Andre-Marie Ampere, a French physicist and mathematician.

Electric Current + + + + Find the rate at which charges are flowing through a cross sectional area of A. elementary charge volume that charge moves through charge increment number of elementary charges distance that charge moves through

Electric Current + + + + This results in another definition for electric current. And current density, J, which is the amount of current per area:

1 How long will it take for 400.0 C of electric charge to pass through a copper wire if the current is measured to be 1.50 A? A 0.00375 s B 267 s C 534 s Answer D 600 s E 634 s

2 How many charge carriers are required to produce 1.50 A of current in a wire of radius 2.00 cm when the carriers have a drift velocity of 1.50 x 10-3 m/s? A 6.47 x 10 23 B 9.87 x 10 23 C 3.36 x 10 24 Answer D 4.97 x 10 24 E 9.46 x 10 24

Resistance and Resistivity Return to Table of Contents

Resistivity When a current is passing through a wire there is a resistance to the motion of the charges, as they bounce off other charges and atoms in the conductor. The resistance is directly proportional to the Electric Field created by the Electric Potential difference and inversely proportional to the current density. Materials have a unique value for the resistivity, ρ, with units of ohm-meters.

Resistivity The resistivity of a material is affected by its temperature. At higher temperatures, resistivity in a metal conductor increases. because as the material heats up, the particles within it begin to vibrate more. As a result the moving charged particles will collide more frequently with the ions decreasing the drift velocity, which decreases the current density. This equation is only valid up to about 373 K.

Ohm's Law We will use this micro property of the material of the conductor to derive the resistance of a conductor's physical dimensions to current flow. For a conductor of length l, assume that the Electric Field created by the Electric Potential is uniform: A is the cross sectional area of the conductor

Define Resistance as: Resistance and Ohm's Law Which leads to Ohm's Law: Resistance has the unit of ohm, Ω, which is volt/amp. Frequently it is written just as V = IR. Not all materials are described by Ohm's Law - the ones that do are called "ohmic" materials. There is not a linear relationship between Electric Potential and Current in semiconductors. Semiconductors are not ohmic materials.

3 What is the resistance of a copper wire with a length of 2.0 m, a radius of 5.0 cm and a resistivity of 1.7 x 10-8 Ω -m? A 9.6 x 10-7 Ω B 1.8 x 10-6 Ω C 2.6 x 10-6 Ω Answer D 3.8 x 10-6 Ω E 4.3 x 10-6 Ω

4 What is the resistance of a gold wire with a length of 0.50 m, a diameter of 6.0 cm and a resistivity of 2.8 x 10-8 Ω -m? A 1.2 x 10-6 Ω B 1.8 x 10-6 Ω C 5.0 x 10-6 Ω Answer D 6.1 x 10-6 Ω E 8.9 x 10-6 Ω

Electromotive Force (EMF) Return to Table of Contents

Electromotive Force A battery converts chemical energy into Electric Potential Energy, which will do work on a charge, increasing its energy and setting it in motion out of the positive terminal, heading for the negative terminal of the battery. As the charge goes through the circuit, it transfers its potential energy through the resistance in the wire into thermal energy. For DC circuits (the current only flows one way), it takes a while for an individual charge to get back to the battery (drift velocity is so small), but it "bumps" the charges in front of it, setting up a chain reaction of moving charges. This is analogous to the flow of water drops in a hose. Of course, its really the electrons that are moving about the circuit, but we're using the definition of conventional current here.

Electromotive Force When the charge gets back, it has a much lower potential energy than when it started its journey. The battery then gives it another boost of energy and it goes again. This is current. The Electric Field that is created by the difference in Electric Potential (The Electric Potential Energy per charge) between the two terminals of the battery propagates at nearly the speed of light (dependent on the conductor properties), while the electrons are moving much slower. When a light that is in a battery circuit is turned on, the light turns on almost instantaneously because it is not waiting for the charge from the battery terminal to get to the light.

Electromotive Force This Electric Potential Energy per charge is called Electromotive Force, or EMF or ε. This is clearly not a force, but the name is preserved for historical reasons. The full EMF is not always delivered to the circuit, since there is an internal resistance present within the battery. This is equivalent to when you studied mechanics, and friction would impact the acceleration of an object. In Electricity, resistance is analagous to friction. Model the battery as shown below: I An "ideal" battery would not have an internal resistance. r R I

Electromotive Force The physical battery is shown within the rectangle and includes the source of the EMF, ε, and r, which represents the internal resistance due to the materials and the chemical reaction occuring within the battery. r R If the switch is open, then no current is flowing, and a Voltmeter measuring the voltage across points a and b would read equal ε. This is called the terminal voltage.

Electromotive Force When the switch is closed, a current flows. This results in the terminal voltage being less than ε, and the current delivered to the circuit being less then ε/r, as one would have expected with a perfect battery. The internal resistance increases as the battery gets older and in cold weather. Both cases contribute to car batteries not being able to start a car in the winter. I r R I

Circuit Components Conducting Wire Resistor Ideal Battery (Source of EMF) Battery (Source of EMF with an internal resistance) A Ammeter V Voltmeter

5 A 9.0 V battery with an internal resistance of 1.5 Ω is connected in series to a light bulb with a resistance of 12 Ω. What is the current in the circuit? A 0.067 A B 0.67 A C 0.71 A Answer D 0.75 A E 6.0 A

6 A 9.0 V battery with an internal resistance of 1.5 Ω is connected in series to a light bulb with a resistance of 12 Ω. What is the terminal voltage of the battery when current is flowing? A 0.08V B 0.96 V C 8.0 V Answer D 9.0 V E 10 V

Energy and Power Return to Table of Contents

Electrical Energy and Power A resistor's thermal energy increases when a current passes through it. The free electrons moving through the resistor bounce off each other and the atoms, increasing the velocity of the electrons and atoms. This increases the thermal energy, also of the resistor - the electrical energy provided by the battery is transferred into kinetic energy and then thermal energy of the resistor. The resistor transfers the increased thermal energy to the environment via conduction, conduction and radiation. The rate at which this energy is transferred is called power (like in mechanics).

Electrical Energy and Power Calculate the electrical energy delivered to the resistor and the rate at which it is transferred to the environment. The battery delivers an Electric Potential Energy of ΔU=QΔV to each charge Q. This Potential Energy is then transferred to the thermal energy released by the resistor. To find the rate of this energy transfer (power), take the derivative with respect to time of the potential energy: ΔV is constant Using Ohm's Law to find two more expressions for power

Electrical Energy and Power The unit of Electrical Power is the Watt (Joule/sec). Just like Mechanical Power. It is fitting that the Watt is used to measure both Mechanical and Electrical Power, since James Watt (Scottish engineer) made substantial improvements to the steam engine. When the steam engine was coupled with a turbine generator, the system would be used to convert mechanical energy to electrical energy. replace variables with their units

7 A toy car's electric motor has a resistance of 17 Ω. What is the power delivered to the motor by a 6.0 V battery? A 0.36 W B 2.1 W C 2.8 W D 28 W Answer E 102 W

8 What current needs to flow through a 450 Ω resistor in order for it to transfer 120 W of power? A 0.14 A B 0.27 A C 0.52 A D 3.8 A Answer E 38 A

Resistors in Series and in Parallel Return to Table of Contents

Resistors in Series Resistors are in a series configuration when they are lined up as above. Because of the conservation of charge, the same amount of charge must pass through every point in the circuit per unit time. Charge is neither created nor destroyed. The rate of charge per unit time is current, thus:

Resistors in Series Let's move to the conservation of energy now - the sum of the voltage drops (energy per unit charge) across the resistors must equal the electric potential provided by the battery.

Resistors in Series Put these equations together, with Ohm's Law for each resistor. The three physical resistors are replaced with one virtual resistor, or equivalent resistor which acts as three resistors in the circuit.

Resistors in Parallel Resistors are in parallel when they have the same electric potential (voltage drop) across each other. There will be a different current through each one, but the three currents will sum up to the current coming out of the battery due to conservation of charge.

Resistors in Parallel Use Ohm's Law again for each current. where

9 What is the equivalent resistance of the below circuit? A 1 Ω B 2 Ω R 1 = 5Ω R 2 = 3Ω C 3 Ω D 8 Ω E 9 Ω V = 9 V Answer

10 What is the current at any point of the below circuit? A 0.89 A B 1.1 A R 1 = 5Ω R 2 = 3Ω C 2.2 A D 8 A E 9 A V = 9 V Answer

11 What is the voltage drop across R 1? A 1.1 V B 3.3 V R 1 = 5Ω R 2 = 3Ω C 5.5 V D 8.2 V E 9 V V = 9 V Answer

12 What is the voltage drop across R 2? A 1.1 V B 3.3 V R 1 = 5Ω R 2 = 3Ω C 5.5 V D 8.2 V E 9 V V = 9 V Answer

13 What is the equivalent resistance of the below circuit? A 1 Ω R 1 = 3Ω B 2 Ω C 3 Ω D 8 Ω R 2 = 6Ω Answer E 9 Ω V = 18V

14 What is the current leaving the battery? A 1 A R 1 = 3Ω B 2 A C 3 A D 8 A R 2 = 6Ω Answer E 9 A V = 18V

15 What is the current passing through R 1? A 1 A R 1 = 3Ω B 2 A C 3 A D 6 A R 2 = 6Ω Answer E 9 A V = 18V

16 What is the current passing through R 2? A 1 A R 1 = 3Ω B 2 A C 3 A D 6 A R 2 = 6Ω Answer E 9 A V = 18V

17 What is the Voltage drop across R 1? A 1 V R 1 = 3Ω B 3 V C 6 V D 9 V R 2 = 6Ω Answer E 18 V V = 18V

18 What is the Voltage drop across R 2? A 1 V R 1 = 3Ω B 3 V C 6 V D 9 V R 2 = 6Ω Answer E 18 V V = 18V

Kirchoff's Rules Return to Table of Contents

Kirchhoff's Rules Circuits are way more complex than just a couple of resistors in series or in parallel. If the circuit is easy enough to analyze by replacing a group of resistors in a perfect parallel combination with an equivalent resistance, or a group in a perfect series combination, then by iteration, the equivalent resistance of the circuit can be calculated. For more complex combinations of circuit elements, Kirchoff's Rules are required. For now, we'll only deal with resistors and batteries - but this also applies to circuits with capacitors and inductors. There are two rules: Junction rule and Loop rule.

Kirchoff's Junction Rule The algebraic sum of the currents into any junction is zero. Or, the amount of current that flows into a junction is the same as the amount of current that flows out of the junction. This is a consequence of the conservation of charge. Junction

Kirchhoff's Loop Rule The algebraic sum of the potential differences in any loop, including those associated with EMFs, resistors, capacitors and inductors equal zero. Conservation of Energy is the source of this rule. R 3 R 1 R 2 R 4 ΔV 1 ΔV 2 Can the circuit above be analyzed by iterating resistors in series and parallel and coming up with one equivalent resistance for the circuit?

Kirchhoff's Loop Rule No. The battery at ΔV 2 makes it impossible. If not for that battery, you could combine R 3 and R 4 into one resistor by using the resistors in series rule. Then combine R 1 and R 2 into one resistor by using the resistors in parallel rule. You're left with two resistors in series which are then added together to get a final equivalent resistance. R 3 R 1 R 2 R 4 ΔV 1 ΔV 2 But, Kirchoff's Rules can be used to analyze the circuit.

Kirchhoff's Loop Rule The values of the resistors and the two batteries are all given. The goal is to find the current in each "loop" of the circuit, using the Loop rule and the Junction rule. Start by drawing two loops, and assign a current to each one. Practice will help you choose the right loops. Another loop that could be used is one that goes through both batteries and R 1, R 3, and R 4. R 3 I 1 R I2 1 R 2 R 4 ΔV 1 ΔV 2 The loop directions are not critical. We'll see why a little later.

Kirchhoff's Rules Traverse each loop with your pencil or pen, and account for the ΔV at each circuit element. If the drawn current loop is in the same direction as your traversing, assign a negative value for the potential drop. If the drawn current is opposite the direction of your traversing, assign a positive value for ΔV. When crossing a battery, if you leave the positive terminal, assign a positive ΔV. If you leave the negative terminal, assign a negative ΔV. Add the currents at the junction points using the junction rule. R 3 I 1 I2 Now, move to the next slide and try these rules out. R 1 R 2 R 4 ΔV 1 ΔV 2

Kirchhoff's Loop Rules R 3 I 1 R 1 R I2 2 R 4 ΔV 1 ΔV 2 Practice on this diagram, and create ΔV = 0 equations for each loop. Move on to the next slide when you're done. Be careful with the current through resistor R 2.

Kirchhoff's Loop Rule J I 1 I 2 R 3 I 1 -I 2 R 1 I 1 R 2 I2 R4 ΔV 1 ΔV 2 The current entering junction, J, is I 1, and since I 2 is leaving, heading for R 3, the current going down through R 2 is equal to I 1 - I 2 by the junction rule. Now, it's time for the loop rule.

Kirchhoff's Loop Rule J I 1 I 2 R 3 I 1 -I 2 R 1 I 1 R 2 I2 R4 ΔV 1 ΔV 2 Loop 1: Loop 2:

Kirchhoff's Loop Rule I 1 I 1 J I 2 I 1 -I 2 R 3 R I2 1 R 2 R 4 We now have a system of two equations, and only two variables - I 1 and I 2. With a little algebra, both currents can be solved for. ΔV 1 Loop 1: Loop 2: ΔV 2 If you chose the wrong direction for the currents in the beginning - no problem - the solution will give a negative current - showing its opposite to the initial choice.

Circuit Measuring Devices Return to Table of Contents

D'Arsonval Galvanometer A D'Arsonval Galvnometer can be used to determine the current, electric potential, and the resistance in a circuit based on the interaction of an electric current with a magnetic field. This is an analog device and is rather old fashioned compared to digital ammeters, voltmeters and ohmmeters. A D'Arsonval Galvnometer is comprised of a coil of fine wire that is placed in a permanent magnetic field. A spring is attached to the coil and when a current is present the magnetic field exerts a torque on the coil. The interaction of electricity with magnetism was covered in the Algebra based Physics and AP 2 Physics courses. It will be covered again in more detail in a later unit.

Ammeter An ammeter is a device used to measure the magnitude of the current in a circuit. The current is constant through a circuit element due to the conservation of charge, hence, the ammeter needs to be placed in series with the element that is being measured. This involves breaking the circuit and installing the ammeter in series. A D To measure I net or the current through R 1 an ammeter is placed at points A or D. An ammeter is placed at point B to measure the current through R 2 and at point C to measure the current through R 3. B C

Voltmeter A voltmeter is a device used to measure the potential difference between two points. A voltmeter is placed in parallel with the circuit element to be measured. This uses the concept that elements (including the voltmeter) that are in parallel have the same voltage drop or potential difference. A B C E D F G Because it is placed in parallel, it may be used without interrupting the circuit like an ammeter.

Voltmeter A B C E D F To measure the voltage drop across R 1, connect the voltmeter to points A and B. G To measure the voltage drop across R 2 and R 3, connect the voltmeter to points C and D or points E and F - both will give the same reading as the resistors are in parallel. To measure the voltage drop across the entire circuit, connect the voltmeter to points A and G.

Ohmmeter In order to measure the resistance in a circuit an Ohmmeter, which consists of a battery of known ε, a variable resistor and a galvanometer, is used. The deflection of the galvanometer is dependent on the equivalent resistance of the variable resistor, ε, and the unknown resistor r. Since all quantities are known, the ohmmeter is calibrated to find r. G

Shunt Resistors A key part of the ammeter and voltmeter is the shunt resistor. It enables the measurement of currents and voltages that are higher than the range of the meters. For an ammeter, the shunt resistor is placed in parallel to the galvanometer function that is measuring the current. The voltage drops across the galvanometer and shunt resistor is the same, and by using a small shunt resistor, the galvanometer will receive a smaller value of the current - thus protecting it and enabling it to safely measure the larger current. shunt resistor G leads to circuit to be measured

Shunt Resistors For a voltmeter, the shunt resistor is placed in series with the galvanometer function that is measuring the voltage. The current through the galvanometer and the shunt resistor is the same, and by using a larger shunt resistor, the galvanometer will not interfere with the circuit's current, and also be protected against a larger current and voltage drop. shunt resistor G leads to circuit to be measured

RC Circuits Return to Table of Contents

RC Circuits Resistors in a circuit are fairly easy to analyze. Once a switch is closed, the potential drop across the resistor occurs instantaneously. The same is not true for capacitors. Capacitors in a circuit with resistors are called RC circuits, an example of which is shown below. + - Switch X Y Z The voltage drop across a capacitor does not happen instantaneously.

Charging a Capacitor + - Switch I 0 X Y Z Right after the switch is closed, the current through the circuit is a maximum, I 0 = ΔV/R. There is no charge on the capacitor, so the potential difference across it is zero. The capacitor then starts charging. This produces an electric field that opposes the one created by the battery. This decreases the current through the circuit. The changing current is represented by i. ΔV R = ir for the resistor and ΔV C = q/c for the capacitor, where q is the charge placed on the capacitor by the current, i.

Charging a Capacitor + - Switch I 0 X Y Z At some point, the capacitor is completely charged, the potential difference across the capacitor is equal and opposite to the battery and the current stops flowing. At this time, Q = CΔV, where Q is the maximum charge deposited on the capacitor by this battery.

Charging a Capacitor + - Switch I 0 X Y Z The current through the RC circuit is a maximum when the switch is closed and is equal to zero after some time interval. But what is this time interval and how does the current change? Please to to the next slide for the answer.

Charging a Capacitor + - Switch I 0 X Y Z Use Kirchoff's Loop rule: We have one equation, but two variables, i and q. It's a good thing that i is the time derivative of q.

Charging a Capacitor + - Switch I 0 X Y Z Making the substitution and separating the variables dq and dt: The next step will be to integrate from t o to an arbitrary time t, where the charge increases from q = 0 to q.

Charging a Capacitor + - Switch I 0 X Y Z

Charging a Capacitor + - Switch I 0 Continuing the math: X Y Z Q is the maximum amount of charge that is deposited on the capacitor.

Charging a Capacitor + - Switch I 0 X Y Z Find the current as a function of time by taking the time derivative of the charge. I 0 is the maximum current in the circuit, at t = 0.

Charging a Capacitor Plot the charge on the capacitor and the current through the circuit as a function of time.

Time constant Both the charge and current time dependent equations have the factor, RC. RC is called the Time constant and is denoted by τ. If τ is small, then the charging time of the capacitor is short, and if τ is greater than the charging time is greater. If the resistance is greater, then the current flow is less, and it takes longer to charge the capacitor. If the capacitance is greater, then it can hold more for a given applied potential difference, and it will take longer to charge. The equations match our physics intuition, so that's always good, and they are also shown as:

Time constant At time, t = τ, the charge reaches 63% of its maximum value, and the current decreases to 37% of its maximum value. This is an alternative definition for the time constant.

Discharging a Capacitor + - Switch I 0 + - X Y Z When the capacitor was fully charged, it was left with a negative charge on the plate near point Z and a positive charge on the plate near point y. No current is flowing, but the direction of the charging current is shown (you'll see why shortly). Switch + - X Y Z Open the switch, and remove the battery. The charge on the capacitor has nowhere to go, so it will stay on the capacitor.

Discharging a Capacitor I 0 Switch + - X Y Z Now, close the switch. The capacitor acts like a battery (it has a potential difference) and current will flow - in the opposite direction of the original charging current (which is why we left that current on the previous slide for comparison). The capacitor will have an initial ΔV and a maximum current, I o, at the start. But, unlike a battery, the capacitor cannot create more electric potential energy from chemical reactions. So, the current will decrease as the charges flow. Over time, the charge and the current will decrease to zero.

Discharging a Capacitor Kirchoff's Loop rule will be used to find the current and charge at any time after the switch is closed. because the charge on the capacitor is decreasing separating the variables The next step will be to integrate from t o to an arbitrary time t, where the charge decreases from q = Q to q.

Discharging a Capacitor Take the time derivative of the charge to find the current behavior.

Discharging a Capacitor The negative sign for the current indicates that the discharging current flows opposite to the charging current.

Discharging a Capacitor Plot the charge on the capacitor and the current through the circuit as a function of time.

19 What is the current through the 5 Ω resistor immediately after the switch is closed? A 0 A B 2 A C 4 A D 6 A E 8 A Answer

20 The circuit has been connected for a long time. The charge on the capacitor is: A 60 µc B 20 µc C 2 µc D 40 µc E 30 µc Answer

21 The circuit has been connected for a long time. What is the current through the 5 Ω resistor? A 8 A B 6 A C 4 A D 2 A E 0 A Answer

22 A capacitor is placed in series with a resistor and a battery. Which of the following graphs represents the charge on the capacitor as a function of time? A B C Answer D E