Physics 2B Sample Midterm Exam #1 (49 points) by Todd Sauke (Actual Midterm will be shorter than this.) Question #1. (6 pts) A point charge Q = -800 nC (nanoCoulombs) and two unknown point charges, q1 and q2, are placed as shown in the figure at right. The electric field at the origin O, due to charges Q, q1 and q2, is equal to zero. We want to determine the values of charges q1 and q2. The electric field vector at the origin has two components (x and y). Because the electric field at the origin is zero, both the x and y components are zero. We can consider the x and y components separately. Remember that the y component of the electric field at the origin due to q1 is zero, since it is on the x axis. What is the y component of the E field at the origin due to the charge Q? (Use a trigonometric function to obtain the y component.) _______________________________ N/C 1 point (check the sign!) What must be the y component of the E field at the origin due to q2 (in order to cancel out the E field from Q)? ________________________________N/C 1 point What must the charge q2 be to provide this field? _________________________________nC 1 point (note units!) Working similarly, what must the charge q1 be in order for the E field at the origin to be zero? _________________________________nC 2 points How many excess electrons are there in the charge Q? _________________________________ 1 point Question #2. (3 pts.) As shown in the figure above, a circular plate with radius 1.96 m contains 793 μC (microCoulombs) of a charge uniformly distributed. We want to find the magnitude of the electric field near the plate. First, what is the charge density, σ, of the given charge distribution? _________________________________C/m2 1 point Now, write the equation from the study guide that relates the field near a sheet of charge to the charge density, σ. ______________________________ 1 point What is the magnitude of the electric field near the plate? _______________________________ 1 point Question #3. (2 points) In the figure at right, charge is placed on an irregularly shaped piece of copper, a good electrical conductor. How will the charge be distributed on the object? A) B) C) D) E) With greatest density neat point E on the flat surface. With greatest density near point C on the surface. With gratest density near point D in the interior. Uniformly throughout the volume. Uniformly over the surface. Question #4. (5 pts.) A nonuniform electric field is directed along the x-axis at all points in space. The magnitude of the field varies with x, but not with respect to y or z. The axis of a cylindrical surface, 0.80 m long and 0.20 m in diameter, is aligned parallel to the x-axis. The electric fields E1 and E2, at the ends of the cylindrical surface, have magnitudes of 3000 N/C and 7000 N/C respectively, and are directed as shown in the figure above. What is the electric flux exiting the cylinder's circular surface at the left end? _____________________________N m2/C 1 point (check your signs) What is the electric flux exiting the cylinder's circular surface at the right end? _____________________________N m2/C 1 point (check your signs) What is the angle between the electric field and the side surface of the cylinder? ______________________________radians ½ point What is the electric flux exiting this side surface? ______________________________ ½ point Write the equation relating the total flux exiting a closed surface to the total charge enclosed within that surface. ______________________________________ 1 point What is the charge enclosed by the cylinder? (check your sign!) ___________________________ ______ 1 point (include units for full credit) Question #5. (5 pts.) Point charges, Q1 = +58 nC and Q2 = -90 nC, are placed as shown in the figure at right. An electron is released from rest at point C. (The electron will experience a force to the right, and accelerate in that direction.) We want to find the speed of the electron as it approaches infinity (far away). The force on the electron changes as it moves away, so we can't use constant acceleration equations. Instead let's use the concept of potential energy, together with the Work-Energy theorem. Taking the potential energy of the electron to be zero when it is at infinity (far away), write the equation from the study guide that gives the potential energy of the electron at point C. _____________________________________ 1 point What is the potential energy of the electron at point C? ______________________________ 1 point What is the equation relating the potential energy and the kinetic energy of the electron at t = 0 and at t = ∞ (when the electron is far away)? ____________________________________ 1 point What is the speed of the electron when it approaches infinity (far away)? __________________________________ ______ 2 points (include units to get full credit) Question #6. (2 points) Two conductors are joined by a long copper wire. Thus A) each carries the same free charge. B) the potential on the wire is the average of the potential of each conductor. C) the electric field at the surface of each conductor is the same. D) no free charge can be present on either conductor. E) each conductor must be at the same potential. Question #7. (6 pts.) The voltage and power ratings of a light bulb, which are the normal operating values, are 110 V and 40 W. Assume the filament resistance of the bulb is constant and has a value, R, independent of operating conditions. The light bulb is operated at a reduced voltage, VRed, and the power drawn by the bulb is 25 W. Write the TWO equations relating power, voltage, and resistance for the two cases under consideration (the normal, full-power case, and the special, reduced-voltage case). ______________________________ 1 points _________________________________ 1 points You should have TWO equations and TWO unknowns. What are the TWO unknowns? ___________ 0.5 points ___________ 0.5 points Solve the two equations for the unknown reduced operating voltage VRed. The operating voltage of the bulb is: _________________________ 3 points _______________________________________________________________________________________ Question #8. (4 pts.) Given the same situation as Question #7 above, i.e. a 40 W light bulb, the light bulb is operated with a current which is one half of the current rating of the bulb (one half of the normal current). Write the equation for the reduced electrical power, PRed, as a function of reduced voltage and current (VRed, and IRed). _______________________ ½ point Write the equation for the reduced voltage, VRed, across the resistance of the light bulb as a function of the resistance, R, and the current, IRed, through it. _______________________ ½ point Plug this second equation into the first to obtain a third equation, (eliminating VRed ), and relating the reduced power to the reduced current, IRed, and the resistance, R. ________________________ 1 point The actual power, PRed, drawn by the bulb is: ________________________ 2 points Question #9. (6 pts.) The emf and the internal resistance of a battery are 31 Volts and 2 Ohms, respectively. Draw a figure below showing the equivalent circuit for this battery. Label the positive terminal "a" and the negative terminal "b". 1 point Write an equation for the voltage across the internal resistance. ___________________________ 1 points When the terminal voltage, Vab is equal to 17.4 V, the current through the battery, including direction, is: ________________ Ohms (Circle direction at right:) 2 points Volts (Circle one unit at left, 1 point) Amps Volts per meter from (a to b) OR from (b to a) 1 points _______________________________________________________________________________________ Question #10. (4 pts.) A wire of radius 1.1 mm carries a current of 2.5 A. The potential difference between points on the wire 49 m apart is 2.6 V. Write the equation relating the electric field in the wire to the voltage mentioned in the problem. ______________________________ ½ point What is the electric field in the wire? ________________________________ V / m 1 points Write the equation relating the resistivity of the material of which the wire is made to the resistance, R, of the wire (between the 49 meters separated points) and to the other quantities mentioned in the problem. ____________________________________________ ½ point What is the resistance of the wire (between the 49 meters separated points)? _________________ 1 point What is the resistivity, ρ , of the material of which the wire is made? _____________________________________________ Ω m 1 points Question #11. (6 pts.) A resistor with resistance 470 Ω is in a series with a capacitor of capacitance 8.5 x 10-6 F. We want to determine what capacitance must be placed in parallel with the original capacitance to change the capacitive time constant of the combination to three times its original value. Write an equation for the equivalent capacitance, Ceq, of the parallel combination of the given capacitor, C, and the unknown capacitor, Cu. ______________________________ 1 point Write an equation for the original capacitive time constant, τ0, in terms of the resistance and the given capacitance, C. ______________________________ 1 point Write an equation for the capacitive time constant, τ, in terms of the resistance and the equivalent capacitance of the parallel capacitors, Ceq. ______________________________ 1 point We have these THREE equations. We are told that we want to consider the case for which the time constant, τ, is three times it's original value, τ0. (τ = 3 τ0) This is a FOURTH equation. What are the FOUR unknowns? τ τ0 __________ _________ (given) (given) ________________ 0.5 points _____________ 0.5 points Four equations with four unknowns may seem "hard" or "scary". Of course you should see, however, that this is an easy problem. Most of the equations (like τ = 3 τ0 ) are so easy that normally we wouldn't even count it as an equation (and we wouldn't count τ0 as an unknown). We just plug in the values as we go along with our "real" equations, making substitutions for some unknowns in terms of others for which we have "easy" equations. (A confident, capable, person could actually just write down a single equation, (the one given τ = 3 τ0) and just plug in the values corresponding to the three equations asked for above from their head while writing it all down, keeping only the single unknown, Cu , and solving the one equation with the one unknown.) Solve the equations for the unknown capacitance, Cu , which must be placed in parallel with the original capacitance to change the capacitive time constant of the combination to three times its original value. Cu = ____________________ 2 points

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