Make sure you show all your work and justify your answers in order to get full credit.

Transcription

1 PHYSICS 7B, Lectures & 3 Spring 5 Midterm, C. Bordel Monday, April 6, 5 7pm-9pm Make sure you show all your work and justify your answers in order to get full credit. Problem esistance & current ( pts) Two cylindrical wires of respective resistivities and, respective lengths l and l, and same diameter d are connected to each other, as shown in Figure a. a) Calculate the resistance of the connected wires made of different materials. This can be used to monitor the level of liquid helium in a storage tank. A niobiumtitanium (Nb-Ti) wire of length l spans the entire height of the tank, and an electronic circuit maintains a constant electrical current at all times in the wire. A voltmeter monitors the voltage V across the wire, as shown in Figure b. Because Nb-Ti is superconducting at low temperatures, the portion of the wire immersed in liquid helium (length x) therefore has zero resistivity. The portion above the liquid always has nonzero resistivity. b) Calculate the ratio V/V, where V is the voltage measured when the tank is empty. Give your answer in terms of the fraction f of the tank which is filled with liquid helium. (a) (b) d l l Figure

2 Problem DC circuit (5 pts) Two resistors of resistance and two capacitors of capacitance C are combined as shown in Figure to form a circuit, where the battery sources a voltage E. a) Draw a simplified version of that electrical circuit using only one resistor of equivalent resistance eq and one capacitor of equivalent capacitance Ceq. Express eq and Ceq as a function of and C. b) Before the battery is connected to the circuit, the capacitors are uncharged. Establish the differential equation satisfied by the charge Q accumulating on the equivalent capacitor s plates, using eq and Ceq. You don t need to solve the equation! c) Determine, without any calculation, the current I going through the equivalent circuit immediately after the battery is connected to the circuit, and then after an infinite amount of time. E C C Figure

3 Problem 3 Capacitor & dielectric ( pts) Two coaxial cylindrical conducting shells of identical length L and respective radii and (>) carry uniformly distributed electric charge +Q and Q respectively (Q>). They are separated by a vacuum of permittivity. You may assume that L>>. A cross-section view is presented in Figure 3a. a) Calculate the electric field created at any point M located at a radial distance r from the symmetry axis. Show your work! b) Calculate the absolute value of the voltage between the conducting shells. c) Determine the capacitance of this cylindrical capacitor. d) Calculate the capacitance if the gap between the shells is filled by successive dielectric materials of dielectric constant K and thickness d for the inner one, and dielectric constant K for the outer one, as sketched in Figure 3b. (a) (b) K K d Figure 3: cross-section views Problem 4 Electric potential & potential energy ( pts) An electron of mass m and electric charge -e, initially at rest, is released from infinity along the symmetry axis of a uniformly charged disk of radius. The flat disk carries positive surface charge distribution. Calculate the kinetic energy of the electron at distance z from the center of the disk (Fig.4). You may assume that the gravitational potential energy is negligible. z O Figure 4

4 Problem 5 Electric field & potential (5 pts) We assume that atoms can be modeled by considering a spherical negative charge distribution (r) extending beyond the radial distance a (a >), around the nucleus of charge q (q>). At distance r from the center O of the atom, the electric potential is given by the following expression: q r / a V ( r) e 4 r a) Determine the direction and magnitude of the electric field E (r) created by this charge distribution at a distance r from the origin. b) Calculate the flux (r) Φ E of the electric field through a sphere of center O and radius r. c) Calculate the electric charge Qtot enclosed in the sphere of center O and radius r. What is the limit of Qtot when r? Interpret your result. d) Determine the negative charge distribution (r). Hint: it might be helpful to express the infinitesimal amount of negative charge dq contained in a spherical shell of thickness dr.

5 F = Q Q 4πɛ r ˆr = kq Q r ˆr E = V = F = Q E dq 4πɛ r ˆr = ρ = dq dv σ = dq da λ = dq dl p = Q d τ = p E U = p E Φ E = E d A E d A = Q encl ɛ U = Q V V = kdq ˆr E d l r dq kdq 4πɛ r = r E = V Q = CV C eq = C + C (In parallel) C eq = C + C (In series) ɛ = κɛ C = κc U = Q C ɛ U = E dv I = dq dt V = I = ρ l A ρ(t ) = ρ(t )( + α(t T )) P = IV I = j da j = nq v d = E ρ eq = + (In series) eq = + (In parallel) junction I = V = loop y(t) = B A ( e At ) + y()e At solves dy dt = Ay + B y(t) = y max cos( At + δ) solves d y dt = Ay f = f r ˆr + f r θ ˆθ + f z ẑ d l = drˆr + rdθˆθ + dzẑ (Cylindrical Coordinates) f = f r ˆr + f r θ ˆθ + f r sin(θ) φ ˆφ d l = drˆr + rdθˆθ + r sin(θ)dφ ˆφ (Spherical Coordinates) x n e ax dx = n! a n+ x n e ax dx = (n)! π n! n+ a n+ x n+ e ax dx = n! a n+ ( + x ) / dx = ln(x + + x ) ( + x ) dx = arctan(x) ( + x ) 3/ x dx = + x x + x dx = ln( + x ) x + x dx = + x ( ( x dx = ln tan cos(x) + π ) ) 4 ( ( ) ) x dx = ln tan sin(x) sin(x) x cos(x) x e x + x + x ( + x) α + αx + ln( + x) x x (α )α x sin(x) = sin(x) cos(x) cos(x) = cos (x) sin(a + b) = sin(a) cos(b) + cos(a) sin(b) cos(a + b) = cos(a) cos(b) sin(a) sin(b) + cot (x) = csc (x) + tan (x) = sec (x)