Exam 3 Solutions. 1. Which of the following statements is true about the LR circuit shown?

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1 PHY49 Spring 5 Prof. Darin Acosta Prof. Paul Avery April 4, 5 PHY49, Spring 5 Exa Solutions. Which of the following stateents is true about the LR circuit shown? It is (): () Just after the switch is closed R carries zero current. () Just after the switch is closed the two resistors carry identical current. () Just after the switch is closed the current in R is greater than that in R. (4) A long tie after the switch is closed R carries zero current. (5) If the switch is opened after being closed for a long tie the current in both resistors iediately drops to zero. efore the switch is closed, there is no current flowing through the inductor. When the switch is closed, a current will try to start flowing through the inductor, but by Faraday s law and Lenz s law, the inductor will set up an ef to resist the increasing -field, thus trying to keep the current zero. Another way to see this is that iediately after the switch is closed, the voltage across R is the sae as that of the battery. So the current in the branch containing the inductor is just that for an LR circuit containing only R and L. The expression for that is: R ε t L i = e = when t = R. Suppose that the battery in the previous proble is replaced with an AC source with ε = 4sin πt. Which one of the following equations can possibly ef given by ( ) describe the current in the inductor? Again, the current in the inductor is given by that of a driven LR circuit containing only ε = 4sin πt. R and L, since the voltage across R is given by the tie-varying ef ( ) The current in an inductor for a ostly inductive load (reactance fro any capacitance is

2 PHY49 Spring 5 negligible) always lags the voltage: tanφ =, so the only choice aong the R possibilities that shows this lag and is driven at the sae frequency is: ( πt ) i = sin.4 Note that for this solution, the current peaks at a slightly later tie than the voltage because of the negative sign in front of the.4 phase constant. X L. An RLC series circuit has R=Ω, L=. H and is driven by an AC source with ef ε = sin ( 8πt ). For what capacitance will the current be in phase with the voltage source? The current will be in phase with the voltage when the circuit is driven at the natural oscillation frequency: ωl tanφ = ωc = R ω = LC C = = = µ F ω L 8 π s. H ( ) ( ) 4. At what rate in V/sec ust the potential difference between the plates of a parallelplate capacitor with a µf capacitance be changed to produce a displaceent current of.8 A? The appropriate law is Maxwell s Law on induction: dφe! ds= µε C dφ E id ε = displaceent current Φ E da= electric flux through plate of capacitor E! S To get the electric field fro the voltage, one way is to note that the electric field of a charged plate is given by: σ q CV E = = = ε εa εa Fro this we can solve for the displaceent current:

3 PHY49 Spring 5 dφe d CV dv id ε = ε A = C ε A dv id.8 A = = = 6 4 C F 5. A bea of unpolarized light oving downward along the +z axis is sent through a syste of three polarizing sheets whose polarizing directions ake angles of θ =, θ = 5, and θ = 9 to the y axis. What fraction of the light's initial intensity is transitted? The intensity of unpolarized light that passes through one polaroid filter, no atter what the angle, is always cut in half (the average of cos θ is ½): I = I. Once the light passes through that filter, its polarization is defined by the direction of the filter. The intensity of light passing through the second filter is then given by: " I = Icos ( θ θ) = Icos ( ), since it is the relative angle between the two filter sheets that atters. The intensity of light coing fro the third sheet is thus: " " " I = Icos ( θ θ) = Icos ( 4 ) = Icos ( ) cos ( 4 ) =.6I Note that there is nothing special about the 9 angle that would ake the intensity. It would only happen if the iddle sheet was aligned at. 6. If the agnetic field in a plane electroagnetic wave is given by = sin ( kx +ωt )ˆ z, in SI units, then the electric field is given by: Note that this traveling wave oves in the x ˆ direction. For exaple, to keep the phase constant inside the sine function, x ust decrease for increasing t. We also know that the direction of the wave is given by S= E, which eans that the electric field ust µ point in the yˆ direction with the agnetic field in the ẑ direction. Finally, we know that E the ratio of the electric field to the agnetic field is the speed of light: c =. Putting all that together gives: E = c sin ( kx +ωt)ˆ y

4 PHY49 Spring 5 7. The diagra shows the passage of a ray of light fro air into a substance X. The listed angles are easured in degrees. The index of refraction of X is: Snell s Law applies to the angles with respect to the noral to the surface: nair sin 4 = nx sin " sin 4 nx = =.9 " sin " " since the index of refraction of air is very nearly.. 8. An electrically neutral object that is paraagnetic is brought toward one of the open ends of a solenoid agnet (that is switched on). Which of the following is true: () A force will pull the object into the solenoid () A force will repel the object away fro the solenoid () No force is exerted on the object since it is neutral (4) The object will levitate above the solenoid if the solenoid axis is oriented vertically (5) The object will exhibit hysteresis An object that exhibits paraagnetis is one which develops agnetization in the sae direction as an applied agnetic field (the atoic agnetic dipoles line up). So if the object is brought toward the open end of the solenoid, the agnetization is in the sae direction as the field lines. The force on such an object is given by: Fz = µ z z If the open end has pointing in the z direction, and the field is increasing in the z direction (because the open end of the solenoid has a non-unifor field, with increasing toward the inside of the solenoid), the force is in the z direction. This eans that the object will tend to be pulled into the solenoid agnet where the field is increasing. The answer is (). 4

5 PHY49 Spring 5 9. A rectangular loop of wire sits in the x-y plane in a region where there is a unifor agnetic field of = ˆ z, where =.5 T. Two of the sides of the loop are parallel to the y-axis and have a fixed length l y =5 c. The reaining two sides are parallel to the x- axis and have a tie-dependent length given by l x = a + vt, where a=5 c and v=5 c/s. What is the agnitude of the induced ef in the wire loop at tie t=5 s? The appropriate physical law is Faraday s Law of induction: dφ ε = Φ = A= agnetic flux da d ε = = x( a vt) xv # + = # ε =.5 T.5.5 /s = 6.5 V ( )( )( ). A charged capacitor is connected across an inductor to for an LC circuit. When the charge on the capacitor is.5 C the current is 4 A. If the axiu current is 5 A, what is the period of LC oscillations in seconds? q In an LC circuit, the total energy ust reain constant: U = Li + C The subscript denotes the values for the current and charge at the first tie. At a later tie, all the energy is in the inductor (when there is axiu current), so: U = Liax Equating these energies gives us: q U = Li + = Li C q iax i = LC ax q T LC = = = i i π ax ω q T = π =.5 s i ax i 5