Physics 122B Electricity and Magnetism Lecture 10 (Knight: 29.1 to 29.4) Potential and Potential Energy April 18, 2007 Martin Savage

Lecture 9 Announcements Lecture HW Assignments #3 and #4 has been posted on the Tycho system. Assignment #3 is due at 10 PM tonight, and Assignment #4 is due at 10 PM next Wednesday. 9/22/2018 Physics 122B - Lecture 10

Kirchhoff’s Junction Law The current density is generally not the same at all points of the wire 9/22/2018 Physics 122B - Lecture 10

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. 9/22/2018 Physics 122B - Lecture 10

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 9/22/2018 Physics 122B - Lecture 10

Example: The Electric Field in a Wire A 2.0 mm diameter aluminum wire carries a current of 800 mA. What is the electric field strength inside the wire? The electric field strength is 9/22/2018 Physics 122B - Lecture 10

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 9/22/2018 Physics 122B - Lecture 10

Heike Kamerlingh Onnes Superconductivity Heike Kamerlingh Onnes (1853-1926) The “classical” physics we are studying is an approximation to quantum mechanics. In the quantum domain, under certain circumstances (low temperature, electron pairing) , there may be minimum amount of energy that an electron can lose in a collision. If the probable energy loss falls below that minimum, the system may become a “superconductor”, a material in which the electrical resistance of the material vanishes. Superconductivity was discovered in 1911 by the Dutch physicist Heike Kamerlingh Onnes. Most superconductors exist at only very low temperatures (<20 K), but in 1986 a new class of “warm” superconductors was discovered that maintain their superconducting properties up to 125 K. A current initiated in such a material persists, because there is no electrical resistance to dissipate the energy. e.g. a superconducting ring 9/22/2018 Physics 122B - Lecture 10

Work and Potential Energy Recall from Physics 121 that Emech= K + U is a conserved quantity for particles that interact via conservative forces and that for changes, DEmech = DK + DU = 0. The change in potential energy is: DU = Uf – Ui = -Winteraction forces. If a particle moves a distance Dr while a constant force F is acting on it, then the work done is: W = F·Dr = F Dr cos(q), where q is the angle between the force F and displacement Dr. There are three special cases: q=00, q=900, and q=1800. If the force is not constant, the work is: 9/22/2018 Physics 122B - Lecture 10

The Potential Energy in Two Uniform Fields The gravitational field g near the surface of the Earth is uniform. If a particle moves downward from yi to yf, the gravitational field will do a positive amount of work: Therefore: Similarly, for displacements s in a uniform electric field E, with s parallel to E: 9/22/2018 Physics 122B - Lecture 10

Charges in an E Field One difference between a gravity field g and an electric field E is that a mass m interacting with g is always positive, while a charge q interacting with E may be either positive or negative. However, this is not a problem. A positive charge gains energy as it moves away from the positive plate of a parallel plate capacitor, while a negative charge gains energy as it moves away from the negative plate of the capacitor. In either case, the charge gains kinetic energy as its potential energy decreases. 9/22/2018 Physics 122B - Lecture 10

Example: Conservation of Energy inside a Capacitor A 2.0 cm x 2.0 cm parallel plate capacitor with a 2.0 mm gap is charged to ±1.0 nC. First a proton, and then an electron, are released at the midpoint of the capacitor. What is each particle’s change in Uelec from its release to its collision with a plate? What is each particle’s kinetic energy as it reaches the plate? 9/22/2018 Physics 122B - Lecture 10

Question 1 The electric field of a positively charged rod (end view shown) causes a negative particle to orbit the rod in a closed circular path, as shown. What is the sign of the work done on the charged particle by the electric field of the rod? (A) positive; (B) zero; (C) negative; (D) not enough information to tell. 9/22/2018 Physics 122B - Lecture 10

Review of Mass & Spring The energy of a mass-spring system alternates between potential energy Usp stored in the spring and kinetic energy K residing in the moving mass. An energy diagram shows the energy balance vs. position. 9/22/2018 Physics 122B - Lecture 10

Potential Energy of Point Charges The same approach can be applied to the interaction between two charged particles. Consider the work by particle 1 on particle 2 as it moves from xi to xf: 9/22/2018 Physics 122B - Lecture 10

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. 9/22/2018 Physics 122B - Lecture 10

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. 9/22/2018 Physics 122B - Lecture 10

The Zero of Potential Energy We note that the the equation for electric potential energy says that Uelec®±¥ as r®0 and that Uelec®0 as r®¥. This raises the question of the point at which the potential energy should be zero. Only changes in potential energy DU appear in energy equations and have physical consequences. Therefore, the point at which Uelec= 0 is a matter of choice. Two popular choices: Uelec®0 at r®¥ or r “far away”. (2) Uelec=0 at Earth ground or at some other “normal” state used for reference. 9/22/2018 Physics 122B - Lecture 10

Example: Approaching a Charged Sphere A proton is fired from far away at a 1.0 mm diameter glass sphere that has a charge of q=+100 nC. What is the initial speed the proton must have to just reach the surface of the glass? 9/22/2018 Physics 122B - Lecture 10

Example: Escape Velocity An interaction between two elementary particles causes an electron and a positron to be shot out back-to-back with equal speeds. What minimum speed must each particle have when they are 100 fm apart in order to escape each other? 9/22/2018 Physics 122B - Lecture 10

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. 9/22/2018 Physics 122B - Lecture 10

Example: Launching an Electron Three electrons are spaced 1.0 mm apart along a vertical line. The outer two electrons are fixed in position. Is the center electron in a point of stable or unstable equilibrium? If the center electron is displaced horizontally by an infinitesimal distance, what will be its speed when it is very far away? U13 is same before and after 9/22/2018 Physics 122B - Lecture 10

The Potential Energy of a Dipole A dipole with dipole moment p=qd is in a uniform electric field. We are interested in the work done by the electrical forces as the dipole rotates by an angle f. Note that when the dipole rotates by an infinitesimal angle df, the charge is displaced by ds=r df=(½d)df. 9/22/2018 Physics 122B - Lecture 10

Example: Rotating a Molecule Water molecules have a permanent electric dipole moment of p = 6.2 x 10-30 C m. A water molecule is aligned in an electric field of E = 1.0 x 107 N/C. How much energy is needed to rotate the molecule by 900? 9/22/2018 Physics 122B - Lecture 10

End of Lecture 10 Before the next lecture, read Knight, Chapters 29.5 through 29.7. Lecture HW Assignments #3 and #4 has been posted on the Tycho system. Assignment #3 is due tonite and #4 is due next week. 9/22/2018 Physics 122B - Lecture 10