Physics 122B Electricity and Magnetism Lecture 16 Review and Extension May 2, 2007 Martin Savage

Lecture 16 Announcements Lecture HW Assignment #5 is due by 10pm this evening. Lecture HW Assignment #6 is posted on Tycho and is due next Wednesday at 10pm. The second midterm exam is this Friday. It will cover everything up to the end of Wednesdays lecture, emphasizing the most recent material, but assumes understanding of all material inclusive. 11/7/2018 Physics 122B - Lecture 16

Current and Drift Velocity If the electrons have an average drift speed vd, then on the average in a time interval Dt they would travel a distance Dx in the wire, where Dx = vd Dt. If the wire has cross sectional area A and there are n electrons per unit volume in the wire, then the number of electrons moving through the cross sectional area in time Dt is Ne = n A Dx = n A vd Dt = i Dt . Therefore, This table gives n for various metals. 11/7/2018 Physics 122B - Lecture 16

A Puzzle We discharge a capacitor that has been given a charge of Q = 16 nC, using a copper wire that is 2 mm in diameter and has a length of L = 20 cm. Assume that the electron drift speed is vd = 10-4 m/s. How long does it take to discharge the capacitor? (Note that L/vd = 0.2m/10-4 m/s = 2000 s = 33.3 min.) Points to consider: The wire is already full of electrons. The wire contains about 5x1022 conduction electrons. Q = 16 nC requires about 1011 electrons. A length L’ of wire that holds 16 nC of conduction electrons is 4x10-13 m. L’/vd = 4x10-9 s = 4 ns. That is roughly the discharge time. 11/7/2018 Physics 122B - Lecture 16

Establishing the Electric Field in a Wire (2) The figure shows the region of the wire near the neutral midpoint. The surface charge rings become more positive to the left and more negative to the right. In Chapter 26, we found that a ring of charge makes an on-axis E field that: Points away from a positive ring and toward a negative ring; Is proportional to the net charge of the ring; Decreases with distance from the ring. The non-uniform surface charge distribution creates an E field inside the wire. This pushes the electron current through the wire 11/7/2018 Physics 122B - Lecture 16

A Model of Conduction (1) Now turn on an E field. The straight-line trajectories become parabolic, and because of the curvature, the electrons begin to drift in the direction opposite E, i.e., “downhill”. ax=F/m=eE/m so vx=vix+ axDt = vix+ Dt eE/m This acceleration increases an electrons kinetic energy until the next collision, a “friction” that heats the wire….energy is imparted to the atoms of the lattice. 11/7/2018 Physics 122B - Lecture 16

The Current Density in a Wire Example: A current of 1.0 A passes through a 1.0 mm diameter aluminum wire. What is the drift speed of the electrons in the wire? 11/7/2018 Physics 122B - Lecture 16

Kirchhoff’s Junction Law 11/7/2018 Physics 122B - Lecture 16

Conductivity and Resistivity The current density J = nevd is directly proportional to the electron drift speed vd. Our microscopic conduction model gives vd = etE/m, where t is the mean time between collisions. Therefore: The quantity ne2t/m depends only on the properties of the conducting material, and is independent of how much current density J is flowing. This suggests a definition: J = s E This result is fundamental and tells us three things: Current is caused by an E-field exerting forces on charge carriers; (2) Current density J and current I=JA depends linearly on E; (3) Current density J also depends linearly on s. Different materials have different s values because n and t vary with material type. 11/7/2018 Physics 122B - Lecture 16

Resistivity and Conducting Materials For many applications, it is more convenient to use inverse of conductivity, which is called the resistivity, denoted by the symbol r: r = s = Thus, the current density is J = Es = E/r. Here are the conducting properties of common materials: Units of resistivity are W m Units: ohms = W = Nm2/CA = Nm2s/C2 11/7/2018 Physics 122B - Lecture 16

Resistors and Resistance Conducting material that carries current along its length can form a resistor, a circuit element characterized by an electrical resistance R: R ≡ rL/A where L is the length of the conductor and A is its cross sectional area. R has units of ohms ( W ). Multiple resistors may be combined in series, where resistances add, or in parallel, where inverse resistances add. Rnet Rnet I For identical resistors can simply add the areas For identical resistors can simply add the lengths 11/7/2018 Physics 122B - Lecture 16

The Potential Energy of Like and Unlike Charge Pairs This approach can be applied to pairs of electrically charged particles, whether they have the same or opposite charges. However, for like-sign particles (a) the system energy is positive and decreases with separation, while for opposite-sign particles (b) the system is typically “bound”, so that the net energy is negative and increases (closer to zero) with increasing separation. 11/7/2018 Physics 122B - Lecture 16

The Electric Force as a Conservative Force The electrical force is a “conservative force”, in that the amount of energy involved in moving from point i to point f is independent of the path taken. This can be demonstrated in the field of a single point charge by observing that tangential paths involve no change in energy (because r is constant). Therefore, an arbitrary path can be approximated by a succession of radial and tangential segments, and the tangential segments eliminated. What remains is a straight line path from the initial to the final position of the moving charge, indicating a net work that will be the same for all possible paths. 11/7/2018 Physics 122B - Lecture 16

Multiple Point Charges We have established that both energy and electrical forces obey the principle of superposition, i.e., they can be added linearly without “cross terms”. Therefore, for multiple point charges: Here, “i<j” means that for summing over N particles, the sum over i runs from 1 to N, and the sum over j runs from i+1 to N for each value of i. This it a mathematical trick to avoid counting pairs of point charges twice or having i=j terms, which would give a zero in the denominator. 11/7/2018 Physics 122B - Lecture 16

The Electric Potential In Chapter 25 we introduced the concept of an electric field E, which can be though of as a normalized force, i.e., E = F/q, the field E that would produce a force F on some test charge q. We can similarly define the electric potential V as a charge-normalized potential energy, i.e., V=Uelec/q, the electric potential V that would give a test charge q an electric potential energy Uelec because it is in the field of some other source charges. We define the unit of electric potential as the volt: 1 volt = 1 V = 1 J/C = 1 Nm/C. Other units are: kV=103 V, mV=10-3 V, and mV=10-6 V. Example: A D-cell battery has a potential of 1.5 V between its terminals. 11/7/2018 Physics 122B - Lecture 16

The Electric Potential Inside a Parallel Plate Capacitor Consider a parallel-plate capacitor with (with U0=0) 11/7/2018 Physics 122B - Lecture 16

Graphical Representations of Electric Potential Distance from + plate This linear relation can be represented as a graph, a set of equipotential surfaces, a contour plot, or a 3-D elevation graph. 11/7/2018 Physics 122B - Lecture 16

Field Lines and Contour Lines Field lines and equipotential contour lines are the most widely used representations to simultaneously show the E field and the electric potential. The figure shows the field lines and equipotential contours for a parallel plate capacitor. Remember that for both the field lines and contours , their spacing, etc, is a matter of choice. 11/7/2018 Physics 122B - Lecture 16

Rules for Equipotentials Equipotentials never intersect other equipotentials. (Why?) The surface of any static conductor is an equipotential surface. The conductor volume is all at the same potential. Field line cross equipotential surfaces at right angles. (Why?) Dense equipotentials indicate a strong electric field. The potential V decreases in the direction in which the electric field E points, i.e., energetically “downhill” for a + charge For any system with a net charge, the equipotential surfaces become spheres at large distances. 11/7/2018 Physics 122B - Lecture 16

Visualizing the Potential of a Point Charge The potential of a point charge can be represented as a graph, a set of equipotential surfaces, a contour map, or a 3-D elevation graph. Usually it is represented by a graph or a contour map, possibly with field lines. + Spherical Shells 11/7/2018 Physics 122B - Lecture 16

The Electric Potential of Many Charges The principle of superposition allows us to calculate the potentials created by many point charges and then add the up. Since the potential V is a scalar quantity, the superposition of potentials is simpler than the superposition of fields. 11/7/2018 Physics 122B - Lecture 16

Example: The Potential of Two Charges What is the potential at point p? Note that: 1/4pe0 = 9.0 x 109 Nm2/C2 = 9.0 x 109 Vm/C, which, for problems like this, are more convenient units. 11/7/2018 Physics 122B - Lecture 16

Potential of a Disk of Charge 11/7/2018 Physics 122B - Lecture 16

Example: The Potential of a Dime + + + + + + A dime (diameter 17.5 mm) is given a charge of Q=+5.0 nC. What is the potential of the dime at its surface? What is the potential energy Ue of an electron 1.0 cm above the dime (on axis)? 11/7/2018 Physics 122B - Lecture 16

Finding E from V In other words, the E field components are determined by how much the potential V changes in the three coordinate directions. 11/7/2018 Physics 122B - Lecture 16

Example: Finding E from the Slope of V An electric potential V in a particular region of space where E is parallel to the x axis is shown in the figure to the right. Draw Ex vs x. 11/7/2018 Physics 122B - Lecture 16

Example: Finding the E-Field from Equipotential Surfaces The figure shows a contour map of a potential. Estimate the strength and direction of the electric field at points 1, 2, and 3. ~ 11/7/2018 Physics 122B - Lecture 16

Kirchhoff’s Loop Law Since the electric field is conservative, any path between points 1 and 2 finds the same potential difference. Any path can be approximated by segments parallel and perpendicular to equipotential surfaces, and the perpendicular segments must cross the same equipotentials. Since a closed loop starts and ends at the same point, the potential around the loop must be zero. This is Kirchhoff’s Loop Law, which we will use later. 11/7/2018 Physics 122B - Lecture 16

A Conductor in Electrostatic Equilibrium A conductor is in electrostatic equilibrium if all charges are at rest and no currents are flowing. In that case, Einside=0. Therefore, all of it is at a single potential: Vinside=constant. Rules for conductor. 11/7/2018 Physics 122B - Lecture 16

Forming a Capacitor Any two conductors can form a capacitor, regardless of their shape. The capacitance depends only on the geometry of the conductors, not on their present charge or potential difference. (In fact, one of the conductors can be moved to infinity, so the capacitance of a single conductor is a meaningful concept.) 11/7/2018 Physics 122B - Lecture 16

Combining Capacitors Parallel: Same DV, but different Qs. Series: Same Q, but different DVs. 11/7/2018 Physics 122B - Lecture 16

Energy Stored in a Capacitor 11/7/2018 Physics 122B - Lecture 16

Energy in the Electric Field Volume of E-field Example: d=1.0 mm, DVC=500 V 11/7/2018 Physics 122B - Lecture 16

Dielectric Materials* There is a class of polarizable dielectric materials that have an important application in the construction of capacitors. In an electric field their dipoles line up, reducing the E field and potential difference and therefore increasing the capacitance: E off E on 11/7/2018 Physics 122B - Lecture 16

Electric Fields and Dielectrics In an external field EO, neutral molecules can polarize. The induced electric field E’ produced by the dipoles will be in the opposite direction from the external field EO. Therefore, in the interior of the slab the resulting field is E = EO-E’. The polarization of the material has the net effect of producing a sheet of positive charge on the right surface and a sheet of negative charge on the left surface, with E’ being the field made by these sheets of charge. 11/7/2018 Physics 122B - Lecture 16

Capacitors and Dielectrics* If a capacitor is connected to a battery, so that it has a charge q, and then a dielectric material of dielectric constant ke is placed in the gap, the potential is unchanged but the charge becomes keq. If a capacitor is given a charge q, and then a dielectric material of dielectric constant ke is placed in the gap, the charge q is unchanged, but the potential drops to V/ke. 11/7/2018 Physics 122B - Lecture 16

Resistors and Ohm’s Law 11/7/2018 Physics 122B - Lecture 16

Ohmic and Non-ohmic Materials Despite its name, Ohm’s Law is not a law of Nature (in the sense of Newton’s Laws). It is a rule about the approximately linear potential-current behavior of some materials under some circumstances. Important non-ohmic devices: Batteries, where DV=E is determined by chemical reactions independent of I; Semiconductors, where I vs. DV can be very nonlinear; Light bulbs, where heating changes R; Capacitors, where the relation between I and DV differs from that of a resistor. 11/7/2018 Physics 122B - Lecture 16

The Ideal Wire Model In considering electric circuits, we will make the following assumptions: Wires have very small resistance, so that we can take Rwire=0 and DVwire=0 in circuits. Any wire connections are ideal. Resistors are poor conductors with constant resistance values from 10 to 108 W. Insulators are ideal non-conductors, with R=∞ and I=0 through the insulator. 11/7/2018 Physics 122B - Lecture 16

Circuit Elements & Diagrams These are some of the symbols we will use to represent objects in circuit diagrams. Other symbols: inductance, transformer, diode, transistor, etc. 11/7/2018 Physics 122B - Lecture 16

Applying Kirchhoff’s Loop Law to Many Loops* i1 – i3 Define a minimum set of current loops. Label all elements. Write a loop equation for each loop. (Battery or 0 = SDV). Solve equations for currents Calculate other variables of interest. R1= R2= i3 i1 Vbat= R4= R3= R5= Loop equations: i2 i1 – i2 11/7/2018 Physics 122B - Lecture 16

Applying Kirchhoff’s Junction Law to Many Junctions* Define a minimum set of junction potentials. You can select one ground point. Label all elements. Write a junction equation for each unknown junction . Solve these equations for the unknown junction potentials. Calculate the other variables of interest. R1= J1 R5= Vbat V1 R2= R3= R4= Junction equations: J2 V=0 V2 11/7/2018 Physics 122B - Lecture 16

Energy and Power (1) Example: A 90 W load resistance is connected across a 120 V battery. How much power is delivered by the battery? 11/7/2018 Physics 122B - Lecture 16

Real Batteries (2) DVbat Question: How can you measure rint? Answer: One (rather brutal) way is to vary an external load resistance R until the potential drop across R is ½E. Then R=rint because each drops ½E. 11/7/2018 Physics 122B - Lecture 16

Voltmeters vs. Ammeters An ideal voltmeter has infinite internal resistance. It must be connected between circuit elements to measure the potential difference between two points in the circuit. V An ideal ammeter has zero internal resistance. It must be inserted by breaking a circuit connection to measure the current flowing through that connection in the circuit. A I X 11/7/2018 Physics 122B - Lecture 16

Example: Analyzing a Complex Circuit (2) 11/7/2018 Physics 122B - Lecture 16

Lecture 16 Announcements Lecture HW Assignment #5 is due by 10pm this evening. Lecture HW Assignment #6 is posted on Tycho and is due next Wednesday at 10pm. The second midterm exam is this Friday. It will cover everything up to the end of Wednesdays lecture, emphasizing the most recent material, but assumes understanding of all material inclusive. 11/7/2018 Physics 122B - Lecture 16