Lesson 4 Circuits and Resistance

Class 9 Today we will: learn about current, voltage, and power in circuits. learn about resistance of materials and how resistance depends on geometry and temperature. introduce Ohm’s law.

Current, Voltage, and Power in Simple Circuits

Current Benjamin Franklin didn’t know if current was caused by positive charges moving or negative charges moving, so he took a guess… and got it wrong. Current Current was defined as the direction positive charge in a wire would travel.

Current In reality, negative charges are moving in the opposite direction to the current. Current However, we usually ignore that and talk about “positive charge carriers” in a wire that move in the direction of the current.

Current Current is the charge that moves past a point in the wire per unit time.

Current Units of current are amperes or amps. Car batteries deliver several hundred amperes. Most electronic circuits run on a few mA.

Voltage Voltage is electric potential in circuits. Voltage is provided by batteries or generators. A battery pushes electrons out the negative terminal and sucks electrons into the positive terminal.

Voltage If we attach a wire to the positive terminal, a few go into the battery, leaving a positive charge on the surface of the wire. The electrons stop moving when the surface charge on the wire pulls the electrons in the wire with the same force as the positive charge on the battery. + + + + + + + + + + + + +

Circuits If we connect a wire from the positive to the negative terminal of the battery, current will continue to flow through the circuit. – – – – + + + + – + – – – – + ++ + + +

Circuits Charge remains on the surface of the wire. The surface charge is positive near the positive terninal and negative near the negative terminal. The charge density is greatest near the terminals of the battery. Current flows uniformly through the entire cross-section of the wire. – – – – + + + + – + – – – – + ++ + + +

Circuits A 10V battery gives 10eV of energy to each electron that passes through it. Collisions with atoms in the wire cause each electron to lose 10eV of energy every time it goes around the circuit. – – – – + + + + – + – – – – + ++ + + +

Ground The ground acts like a huge conductor. Current can flow into the ground or out of the ground without any limits. The two circuits below are equivalent. + +

Circuits Definitions: An open circuit is one where there is an open switch or a broken wire so that no current flows. A closed circuit is one in which there is a continuous path for current to flow from positive to negative. A short circuit is one where there is an unintentional current path to ground. Currents, sometimes large, flow where they should not, leading to shock and fire hazards.

Circuits What good does flowing charge do? Produces heat, light. Produces magnetic fields – used in motors, vibrators, etc. to give mechanical power. Produces electromagnetic radiation – radio waves for communication. Electronics: amplification, logic, light detection, radiation detection, cathode-ray tubes, etc.

Power Each time an electron goes through a battery, it gains energy. The total energy gained per second is: Power provided by the battery! P=iV

Where does the energy go? Electrons collide with other electrons in atoms and quickly reach terminal velocity – so they don’t keep gaining kinetic energy.

Where does the energy go? P=iV In a wire, it goes to heat. In other devices it can go to light, mechanical energy, energy of radiation fields, etc.

Resistance

Resistance in a Wire Definition: In general R is a function of I, and V. For many materials R is nearly a constant. When R is a constant, we call the material “ohmic.”

Resistors Devices to increase the resistance in part of a circuit. Made of graphite chunks, wire wound around a core, etc. Used to Produce heat or light Adjust current flow and voltages in circuits.

Resistors Even if we don’t want the resistance, we often need to account for resistance in cables, electronic devices, etc.

Ohm’s Law We assume resistors have constant V. Resistance has units of ohms, written as an upper case omega. Typical resistances range from a few ohms to several megohms.

Graphite Resistors Resistance is color coded. R=5600 Ω (+/-10%) 1 2 3 1 2 3 4 5 6 7 8 9 5% 10% 20% 5 6 2 10% second digit # of zeros tolerance first digit R=5600 Ω (+/-10%)

What Affects Resistance? Material Length Cross-sectional area Temperature

Resistance and Geometry One block has V, I, R.

Resistance and Geometry Take two blocks with I going through each. Voltage is Current is Resistance is

Resistance and Geometry Take two blocks with I going through each. Voltage is Current is Resistance is

Resistance and Resistivity ρ is the resistivity. It depends on the material from which the resistor is made. The units of resistivity are Ωm. σ = 1/ ρ is the conductivity Head into series and parallel….

Resistance and Temperature We assume that resistance varies linearly with temperature.

Resistance and Temperature If T = T0, then R =R0.

Resistance and Temperature α is the temperature coefficient of resistivity (resistance). α is usually positive. α is negative for graphite.

Class 10 Today we will: learn how to determine if two resistors are in series or parallel. find out how resistors combine when connected in series and parallel. work examples of series-parallel reduction to find current, voltage and power in resistance networks.

Resistors in Series Have the Same Current Take two resistors with I going through each. Voltage is Current is Resistance is

Resistors in Series

Resistors in Parallel Have the Same Voltage Take two blocks with I going through each. Voltage is Current is Resistance is

Resistors in Parallel Have the Same Voltage

A Test for Resistors in Series Look at the wire connecting the two resistors. Is there any junction between the resistors? The resistors are connected in series. The resistors are NOT connected in series. no yes

A Test for Resistors in Parallel Look at the wire connecting one end of the first resistor to one end of the second resistor. Is there a circuit element (a junction is OK and usually there are junctions) along this wire? yes no The resistors are NOT connected in parallel. Look at the wire connecting the other end of the first resistor to the other end of the second resistor. Is there a circuit element along this wire? no yes The resistors are connected n parallel. The resistors are NOT connected n parallel.

Series-Parallel Quiz Answer the following six questions to see if you understand what series and parallel mean.

Resistors A and B are in series parallel neither

Resistors A and B are in series parallel neither

Resistors A and B are in series parallel neither

Resistors A and B are in series parallel neither

Resistors A and B are in series parallel neither

Resistors A and B are in series parallel neither

Quiz Answers 1. series 2. neither 3. neither 4. parallel 5. series 6 Quiz Answers 1. series 2. neither 3. neither 4. parallel 5. series 6. parallel

Series- Parallel Reduction Find a combination in series or parallel. Combine resistors into a single equivalent resistor. Repeat until there is only one resistor. The voltage across the resistor is the same as the voltage across the battery.

Series- Parallel Reduction Find V, I, R, P for the last step. Bootstrap your way back to the beginning, diagram by diagram. What if there are resistors that aren’t in series or parallel? --- You’ll need to use Kirchhoff’s Laws which we’ll learn later.

Now we’ll work some examples…

Find all the currents, voltages, and powers

2A 24V,2A

2A 2A 2A

2A, 4V 2A 2A, 20V

4V 4V 2A 2A, 20V

4V, 2/3 A 4V, 4/3 A 2A 2A, 20V

2/3 A 2/3 A 4V, 4/3 A 2A 2A 2A

2/3 A, 8/3 V 2/3 A, 4/3 V 4V, 4/3 A 2A 2A, 4V 2A, 16V

16/9 W 8/9 W 2/3 A, 8/3 V 2/3 A, 4/3 V 4V, 4/3 A 16/3 W 2A 2A, 4V 48 W 2A, 16V 8 W 32 W

Using Meters Ammeters measure current. they must be paced in series with other circuit elements so current flows through them. Ammeters should have very small voltage. Voltmeters measure voltage. To measure the voltage between two points, you connect the two leads of the meter to those points. Therefore voltmeters are placed in parallel. Voltmeters should have large voltage.

Real Batteries Real batteries have internal resistance. When they are placed in a circuit we can represent them as a resistor in series with an ideal battery. There’s a voltage drop across the internal resistance. This means that the full voltage of the ideal battery isn’t available to the circuit.

Class 11 Today we will: discuss Kirchhoff’s loop and node equations. learn how to determine the number of loop and the number of node equations we will need. write Kirchhoff’s equations for a sample circuit.

Kirchhoff’s Junction Rule Current into a junction equals current out of a junction. Comes from conservation of charge.

Kirchhoff’s Junction Rule It’s like water in pipes – the water flowing into a junction must flow out again.

Kirchhoff’s Loop Rule The net change in voltage around a closed loop is zero. Comes from Conservation of energy

Kirchhoff’s Loop Rule It’s like moving a ball around any closed path, the change in gravitational potential energy is zero.

Applying Kirchhoff’s Laws This is a “turn the crank” approach – but it works well.

Mark each junction. A B D C

Here there are 6 currents. 2. Label all currents – between each pair of junctions. Choose a direction – it doesn’t have to be right! Here there are 6 currents. A I1 I3 B I6 D I4 I5 I2 C

Here there are 4 junctions, so there are 3 junction equations. 3. Number of junction equations: If there are N junctions, there are N-1 junction equations. Here there are 4 junctions, so there are 3 junction equations. A I1 I3 B I6 D I4 I5 I2 C

4. Write the junction equations: Current in = Current out. B I6 D I4 I5 I2 C

Here: 6-3=3, so we need 3 loops. 5. Number of loop equations: Number of currents – Number of junction equations. Here: 6-3=3, so we need 3 loops. A I1 I3 B I6 D I4 I5 I2 C

6. At least one loop must cover every circuit element. 1 2 B I6 D I4 I5 3 I2 C

7. Put a plus or minus on every resistor (side current goes in is +) and battery (+ is positive terminal). A I1 + + I3 + 1 + 2 B I6 D + + I4 I5 3 I2 + + C

8. Write the loop equations. + + I3 + 1 + 2 B I6 D + + I4 I5 3 I2 + + C

8. Write the loop equations. When you go around the loop, ignore the current arrows! Follow the loop in the direction of the loop arrow! A I1 + + I3 + 1 + 2 B I6 D + + I4 I5 3 I2 + + C

8. Solve the system of equations using your favorite method 8. Solve the system of equations using your favorite method. I’ll usually just ask for the equations.