Summary Vector Scalar 2-particle Force: (N) Potential Energy: (J)             1-particle             Electric Field: (N/C) Potential: (V)

Review Current: (A) Batteries Drift velocity and Ohm’s Law: Amount of charge per unit time past a point Driven by difference in potential Direction follows motion of positive charges Batteries Drift velocity and Ohm’s Law:

Review Resistance: (Ω) Measures how well charge flows A geometric property of the resistor Resistance (R) vs. resistivity (ρ)

Capacitors A capacitor uses potential difference to store charge and energy Example: parallel-plate capacitor clouds Section 18.4

Capacitance Defined Capacitance is the amount of charge that can be stored per unit potential difference The capacitance is dependent on the geometry of the capacitor For a parallel-plate capacitor with plate area A and separation distance d, the capacitance is Unit is farad, F 1 F = 1 C/V Section 18.4

Energy in a Capacitor, cont. Storing charge on a capacitor requires energy The total energy stored is equal to the energy required to move all the packets of charge from one plate to the other Section 18.4

Energy in a Capacitor, cont. The total energy corresponds to the area under the ΔV – Q graph Energy = Area = PEcap Q is the final charge ΔV is the final potential difference Section 18.4

Energy in a Capacitor, Final From the definition of capacitance, the energy can be expressed in different forms These expressions are valid for all capacitors Section 18.4

Dielectrics Most real capacitors contain two metal “plates” separated by a thin insulating region Many times these plates are rolled into cylinders The region between the plates typically contains a material called a dielectric Section 18.5

Dielectrics, cont. Inserting the dielectric material between the plates changes the value of the capacitance The change is proportional to the dielectric constant, κ Cvac is the capacitance without the dielectric and Cd is with the dielectric κ is a dimensionless factor Generally, κ > 1, so inserting a dielectric increases the capacitance Section 18.5

Dielectrics, final When the plates of a capacitor are charged, the electric field established extends into the dielectric material Most good dielectrics are highly ionic and lead to a slight change in the charge in the dielectric Since the field decreases, the potential difference decreases and the capacitance increases Section 18.5

Chapter 19b Electric Circuits Circuit Diagrams? Image taken from www.xkcd.com

Parallel and Series Series—the same current passes through each component. Parallel—the same potential change across each component Components can be neither in series nor in parallel Cannot be reduced to equivalence

Equivalence Multiple circuit elements can be replaced by a single equivalent circuit element without affecting the rest of the circuit Equivalence is different for resistors and capacitors Equivalence is different for series and parallel Reducing a circuit to equivalent components often simplifies circuit analysis Always reduce parallel before series

Capacitors in Series For capacitors connected in series: ΔVtotal = ΔV1 + Δv2 + … The equivalent capacitance is The equivalent capacitance is smaller in series than any of the individual capacitors Section 19.5

Capacitors in Parallel For capacitors connected in parallel Qtotal = Q1 + Q2 The equivalent capacitance is The equivalent capacitance is larger in parallel than any of the individual capacitors Section 19.5

Resistors in Series For resistors in series ΔVtotal = ΔV1 + Δv2 + … The equivalent resistance is The equivalent resistance is larger in series than any of the individual resistor Section 19.4

Resistors in Parallel For resistors in series Itotal = I1 + I2 + … The equivalent resistance is The equivalent resistance is smaller in parallel than any of the individual resistor Section 19.4

Batteries in Series Batteries can also be connected in series The combination of two batteries in series is equivalent to a single battery with emf of We will not deal with batteries in parallel Section 19.4

Circuits An electric circuit is a combination of connected elements forming a complete path through which charge is able to move Circuits are represented by circuit diagrams a good approximation is Rwire=0 Since the resistance of the wires is much smaller than that of the resistors Section 19.3

Circuit Symbols Section 19.3

Circuits, cont. Calculating the current in the circuit is called circuit analysis Two types of circuits: DC stands for direct current The current is of constant magnitude and direction at the source AC stands for alternating current (Ch. 22) Source current changes magnitude and/or direction The current can be viewed as the motion of the positive charges traveling through the circuit Section 19.4

Circuits, cont. There must be a complete circuit for charge flow There must be a return path from the resistor for the current to return to the voltage source If the circuit is open, there is no current flow anywhere in the circuit Section 19.4

Circuit Analysis The main tools of circuit analysis are equivalence, Ohm’s Law, and Kirchhoff’s Rules Loop Rule: the change in potential energy of a charge as it travels around a complete circuit loop must be zero Junction Rule: the amount of current entering a junction much be equal to the current leaving it

Kirchhoff’s Loop Rule Conservation of energy is the heart of the Loop Rule Consider the electric potential energy of a test charge moving through the circuit The test charge gained energy when it passed through the battery It lost energy as it passed through the resistor The energy is converted into heat energy inside the resistor Section 19.4

Kirchhoff’s Loop Rule, cont. Since PEelec = q V, the loop rule also means the change in the electric potential around a closed circuit path is zero ΔV = ε – I R = 0 For the entire loop Assumes wires have no resistance Section 19.4

Power In the resistor, the energy decreases by Power is energy per unit time Applying Ohm’s Law, P = I² R = V² / R The circuit converts chemical energy from the battery to heat energy in the resistors Power is only dissipated through resistance Section 19.4

Kirchhoff’s Junction Rule Conservation of charge is the heart of the Junction Rule The points where the currents branch are called junctions or nodes Since charge cannot be created or destroyed, all charge entering a junction must leave the junction Section 19.4

Kirchhoff’s Junction Rule, cont. Since all charge entering a junction must leave the junction, the current entering must equal the current exiting the junction IT = I1 + I2 Both of Kirchhoff’s Rules may be needed to solve a circuit Section 19.4

Direction of Current Current directions are chosen arbitrarily The signs of the potentials in Kirchhoff’s loop rule will depend on assumed current direction Moving parallel to the current produces a potential drop across a resistor The sign of the currents in Kirchhoff’s junction rule will depend on assumed current direction Currents entering a junction are positive, currents leaving are negative Section 19.4

Direction of Current, cont. After solving the equations, the sign of the current indicates the direction if the current is positive the assumed direction of the current is correct If the current is negative, the direction of the current Is opposite the assumed direction Section 19.4

Using Kirchhoff’s Rules In general, if a circuit has N junctions, the junction rule can be used N – 1 times The loop rule can be used as long as it contains at least one circuit element that is not involved in other loops When using the loop rule, pay close attention to the sign of the voltage drop across the circuit element You can go around the loop in any direction Section 19.4

Using Kirchhoff’s Rules, cont. Reduce all equivalences first Apply the junction rule as many times as gives independent equations Use the loop rule to get as many equations as unknowns in the system Section 19.4

Example:

Real Batteries An ideal battery always maintains a constant voltage across its terminals The value of the voltage is the emf of the battery A real battery is equivalent to an ideal battery in series with a resistor, Rbattery This is the internal resistance of the battery Section 19.4

Real Batteries The current through the internal resistance and the external resistor is The potential difference across the real battery’s terminals is Section 19.4

Circuits with Capacitors Kirchhoff’s Rules can be applied to all kinds of circuits RL circuits The change in the potential around the circuit is +ε – I R – q / C = 0 Solving for I shows that I and q will be time dependent Section 19.5

Charging Capacitors When the circuit is open, there is no current in the circuit and no charge on the capacitor When the switch is closed, current carries a positive charge to the top plate of the capacitor When the capacitor plates are charged, there is a nonzero voltage across the capacitor Current can no longer flow Section 19.5

Charging Capacitors, cont. The current in the circuit is described by The voltage across the capacitor is The charge is given by τ = RC and is called the time constant Section 19.5

Time Constant Current Voltage and charge At the end of one time constant, the current has decreased to 37% of its original value At the end of two time constants, the current has decreased to 14% of its original value Voltage and charge At the end of one time constant, the voltage and charge have increased to 63% of their asymptotic values Section 19.5

Charging Capacitors, cont. Just after the switch is closed The charge is very small Vcap is very small I = ε / R Section 19.5

Charging Capacitors, cont. When t is large The charge is very large Vcap ≈ ε The polarity of the capacitor opposes the battery emf The current approaches zero Section 19.5

Discharging the Capacitor Current: Voltage: Vcap = ε e-t/τ Charge: q = C ε e-t/τ Time constant: τ = RC, the same as for charging Section 19.5

Uses of Capacitance

Ammeters An ammeter is a device that measures current An ammeter must be connected in series with the desired circuit branch An ideal ammeter will measure current without changing its value Must have a very low resistance Section 19.6

Voltmeters A voltmeter measures the voltage across a circuit element It must be connected in parallel with the element An ideal voltmeter should measure the voltage without changing its value The voltmeter should have a very high resistance Section 19.6

Temperature Dependence of Resistance As temperature increases, the ions in a metal vibrate with larger amplitudes This causes more frequent collisions and an increase in resistance For many metals near room temperature, ρ = ρo [1 + α(T – To) α is called the temperature coefficient of the resistivity The resistivity and resistance vary linearly with temperature Section 19.10

Superconductivity At very low temperatures, the linearity of resistance breaks down The resistivity of metals approach a nonzero value at very low temperatures In some metals, resistivity drops abruptly and is zero below a critical temperature Metals for which the resistivity goes to zero are called superconductors Section 19.10