Physics 9 Summer 2010 Midterm For the midterm, you may use one sheet of notes with whatever you want to put on it, front and back. Please sit every other seat, and please don t cheat! If something isn t clear, please ask. You may use calculators. All problems are weighted equally. PLEASE BOX YOUR FINAL ANSWERS! You have the full length of the class. If you attach any additional scratch work, then make sure that your name is on every sheet of your work. Good luck! 1. An electron, e, with charge q e and mass m, and a positron e +, with charge +q e and the same mass m, revolve around their common center of mass under the influence of their attractive Coulomb force. (a) Find the speed v of each particle in terms of the charge, mass, ɛ 0, and their separation distance L. (b) How long does it take for each particle to make a complete orbit? (Again, express this time in terms of the same variables above.) v e + e - L v 1
2. A hollow non-conducting sphere of inner radius a and outer radius b carries a uniform charge density, ρ. The charge density is constant in the region a r b, such that the total charge on the sphere is Q. (a) Determine the charge density, ρ, in terms of the total charge and the radii, a and b. (b) What is the magnitude of the electric field inside the hollow region of the sphere, where r a? (c) What is the magnitude of the electric field outside the sphere, where r b? (d) What is the magnitude of the electric field inside the material of the sphere, where a r b? r a b 2
3. Recall that the electric field at a distance x of a ring of charge Q and radius R is given by E = 1 4πɛ 0 Qx (x 2 + R 2 ) 3/2 î. (a) Determine the potential of this ring at the position x, relative to infinity, using the electric field above. R x (b) What is the potential at the center of the ring? (c) Does the potential in part (b) give the correct electric field at the center of the ring? Q 3
4. Suppose you attach a battery, E, to a switch, S, resistor, R, and capacitor, C, in a closed series circuit. (a) Draw this circuit, including all the labels. (b) Using Kirchhoff s loop rules, determine the equation describing the charge in the circuit, after the switch closes at time t = 0. (c) Show that the charge q (t) = Q 0 ( 1 e t/τ ), where Q 0 and τ are constants, satisfies the equation in part (b), and find Q 0 and τ in terms of E, R, and C. (d) In terms of τ, how long does it take, after the switch is closed at t = 0, for the current to drop to 1/e of it s initial value (where e = 2.71828...)? 4
5. Although the evidence is weak, there has been concern in recent years over possible health effects from the magnetic fields generated by the electrical wiring in your home and electrical transmission lines. The magnetic field strength of the Earth is about 50 µt. (a) The current carried by the electrical wiring in the walls in your home rarely exceeds 10 amps. What is the magnetic field strength 1 meter from a long straight wire in your walls carrying this current? (b) What percentage of the Earth s magnetic field is your answer from part (a)? (c) High-voltage transmission lines, on tall towers, typically carry currents of 200 amps, at voltages of up to 500,000 volts. Although this is much larger than household currents, the lines are roughly 20 meters overhead. Estimate the magnetic field strength on the ground underneath such lines. (d) What percentage of the Earth s magnetic field is your answer from part (c)? 5
The potential energy between a pair of neutral atoms or molecules is very well-approximated by the Lennard-Jones Potential, given by the expression [ (σ ) 12 ( σ ) ] 6 P E(r) = 4ɛ, r r where ɛ and σ are constants, and r is the distance between the molecules. The potential energy is plotted in the figure to the right. The vertical axis is in units of ɛ, while the horizontal axis is in units of σ. Extra Credit Question!! The following is worth 10 extra credit points! Energy 8 7 6 5 4 3 2 1 0-2 -3-4 Molcular Bond Energy 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25-1 Distance (a) Why does the potential energy approach zero as the distance gets bigger? (b) At what separation distance, in terms of σ and ɛ, is the potential energy zero? (c) At approximately what distance is the system in equilibrium? What is the potential energy at that distance? (Express your answers in terms of σ and ɛ.) (d) How much energy would you need to add to the system at equilibrium in order to break the molecular bonds holding it together? Why? (e) How much energy is released in the breaking of those molecular bonds? Why? Note - no calculation is needed to answer these problems! 6
Some Useful Constants. Some Possibly Useful Information Coulomb s Law constant k 1 4πɛ 0 = 8.99 10 9 Nm2 C 2. The magnetic permeability constant µ 0 = 4π 10 7 N A 2. Speed of Light c = 2.99 10 8 m/s. 11 Nm2 Newton s Gravitational Constant G = 6.672 10. kg 2 The charge on the proton e = 1.602 10 19 C The mass of the electron, m e = 9.11 10 31 kg. The mass of the proton, m p = 1.673 10 27 kg. Boltzmann s constant, k B = 1.381 10 23 J/K. 1 ev = 1.602 10 19 Joules 1 MeV = 10 6 ev. 1 Å = 10 10 meters. Planck s constant, h = 6.63 10 34 J s = 4.14 10 15 ev s. The reduced Planck s constant, h 2π = 1.05 10 34 J s = 6.58 10 16 ev s. Some Useful Mathematical Ideas. { x n+1 x n n 1, n+1 dx = ln (x) n = 1. dx a = ln ( x + a 2 + x 2). 2 +x 2 x dx = 1 (a 2 +x 2 ) 3/2 a. 2 +x 2 Other Useful Stuff. The force on an object moving in a circle is F = mv2 r. The binomial expansion, (1 + x) n 1 + nx, if x 1. Small angle approximation, sin θ tan θ θ, and cos θ 1 if θ is smaller than 10. 7