AP Physics Assignment - Inductance & Capacitance

AP Physics Assignment – Inductance and Capacitance

Reading College Physics – pp. 509 – 521, Chapter 20

University Physics – Chapters 30, 31, 32, 33

Objectives/HW

	The student will be able to:	HW:
1	State and apply Faraday’s Law and Lenz’s Law and solve magnetic induction problems involving changing magnetic flux, and induced emf or eddy currents.	1 – 16
2	Solve problems involving basic principles of generators, including production of back emf.	17 – 21
3	State and recognize Maxwell’s equations and associate each equation with its implications.	22 – 23
4	Define and calculate capacitance and solve related problems including those that involve parallel or series capacitors.
5	Analyze RC circuits in terms of the appropriate differential equation and resulting exponential functions for charge, current, voltage, etc.
6	Define and calculate inductance and solve related problems including those that involve parallel or series inductors.
7	Analyze RL circuits in terms of the appropriate differential equation and resulting exponential functions for charge, current, voltage, etc.
8	Analyze LC and RLC circuits in terms of the appropriate differential equation and resulting exponential functions for charge, current, voltage, etc.

Homework Problems

1. A permanent bar magnet is located near a circular coil of wire and moves along the axis of the coil as shown in the diagram below. For each of the following events describe the direction of the current induced in the coil. Let a “positive” current mean that current at the top of the coil goes into the page (i.e. the sense of the emf is leftward by the right hand rule). (a) The magnet moves from point A to point B. (b) A force is applied to each side of the coil until it “collapses” into an ellipse with less area than its original circular shape (while magnet remains stationary). (c) The magnet is rotated 90° about its North pole.

2. The square loop of wire in the diagram moves through equally spaced points A through E at a constant velocity. Point C is located at the midpoint between the poles of a horseshoe magnet. (a) Carefully sketch a graph of magnetic flux vs. time for the square loop and label points A through E. (b) Carefully sketch a graph of current vs. time for the square loop and label points A through E. Let positive values represent counterclockwise current in the loop.

3. A circular loop of wire in the xy-plane is near a current along the x-axis as shown below. For each of the following events describe the magnetic flux as increasing, decreasing, or constant and describe the induced current in the loop as clockwise, counterclockwise, or zero. (a) The current increases. (b) The loop is moved in the positive x-direction. (c) The loop is moved in the positive y-direction. (d) The wire is moved in the positive y-direction. (e) The loop is bent and reshaped into an ellipse. (f) The loop is rotated 30° about an axis through its center that is parallel to the y-axis. (g) The loop is rotated 30° about the x-axis.

4. A uniform magnetic field points in the positive z-direction. A wire loop in the shape of a circle with diameter 25.0 cm lies in the xy-plane. The total resistance of the wire is 33 mΩ. Suppose the magnitude of the magnetic field increases at a constant rate from 0.10 T to 0.30 T in 4.0 s. (a) Find the initial magnetic flux for the loop. (b) Find the emf induced in the wire. (c) Find the current in the wire.

5. A single square loop of wire with sides of 5.00 cm is placed in the xy-plane. A uniform magnetic field points in the positive z-direction and has a magnitude that varies as shown in the graph below. (a) Find the induced emf at t = 0.050 s. (b) Find the induced emf at t = 0.15 s. (c) Find the induced emf at t = 0.22 s. (d) Find the induced emf at t = 0.27 s.

6. Suppose the scenario above is repeated with one exception – the width of the square is doubled. (a) What would be the effect on the results? Explain. (b) How would the current induced in the two coils compare? Explain.

7. The magnetic field perpendicular to the plane of a single copper wire loop of diameter 20.0 cm decreases from 0.60 T to zero. The diameter of the wire is 1.30 mm. How much charge moves through the wire as a result of the changing field?

8. A long cylindrical solenoid with radius 2.00 cm and 650 turns per meter is connected to a power supply and the current through it is increased at a constant rate from 0 to 8.00 A in 0.25 s. (a) Sketch a cross-section of the solenoid and the induced electric field that occurs. (b) Find the magnitude of the electric field at r = 1.00 cm from the axis. (c) At what other location(s) would the electric field have the same magnitude as at r = 1.00 cm? (d) The induced electric field does what work on a single electron as it completes a single turn of the wire?

9. Suppose a cylindrical solenoid of radius R and turns per length n has a clockwise current that decreases according to the function I = I₀e^-^kt. Derive expressions for the induced electric field as a function of r, the radial distance from the axis.

10. The diagram below shows a loop of wire with total resistance R and sides of length L. The loop is located in a uniform magnetic field of magnitude B that is confined to the first and second quadrants (drops to zero in other quadrants). (a) Determine an expression for the current induced in the wire if the loop moves downward with constant speed v. (b) Find the amount of force that must be applied to the loop in order to move it thusly. (c) Find the energy dissipated by the electric current and show that is equals the work done to remove the loop from the field.

11. The figure below depicts a circular loop of wire with 330 turns, radius 15.0 cm, and resistance 4.00 Ω held in a horizontal plane. Also shown is the Earth’s magnetic field which points north and down at an angle of 50.0° and has magnitude 55 μT. (a) Find the initial magnetic flux for a single turn of the coil. The coil is now rotated 50.0 ° so that the flux drops to zero. (b) In what direction does current flow? (c) In order to produce a measurable current that averages 10.0 μA, the repositioning of the loop must occur in what amount of time?

12. At a certain location the Earth’s magnetic field has magnitude 0.50 G and points precisely horizontal and North. A car with a vertical antenna of length 80.0 cm travels at 25.0 m/s on a level roadway. (a) In what direction should the car travel if the induced emf in the antenna is to be maximized and positive at the top? (b) What is the maximum induced emf? (c) Repeat, but this time with the antenna tilted at 30.0 ° relative to horizontal.

13. A long metal bar at a construction site rolls southward and down an incline of 30.0° in a region where the Earth’s magnetic field is 0.60 G north and down at an angle of 50.0° relative to horizontal. If bar reaches a speed of 10.0 m/s and has a length of 4.0 m, what is the induced potential difference between its ends? Which end is “positive”?

14. A metallic bar moves along frictionless metallic rails at a constant speed of v = 5.0 m/s as shown in the diagram below. The rails are connected to a resistor of 3.0 Ω. The entire circuit is in a uniform magnetic field of 0.40 T. (a) Find the resulting current in the circuit. (b) Determine the applied force necessary to move the bar in such a fashion. (c) Find the rate at which mechanical work is done in moving the bar. (d) Find the power dissipated by the resistor.

15. A straight piece of wire is moved along a U-shaped piece of wire in a uniform magnetic field of 0.75 T as shown below. All of the wire has resistance per unit length equal to 0.25 Ω/m and the straight wire moves at constant speed 4.0 m/s to the left. Find the current when: (a) x = 7.0 cm and (b) x = 2.0 cm.

16. A square loop of wire with sides of length 5.00 cm moves at 10.0 m/s away from a long wire with current 20.0 A. The square and the wire are in the same plane. For the instant when the center of the square is 10.0 cm from the wire determine: (a) the magnetic flux through the loop and (b) the induced emf.

17. A simple generator consists of a rectangular coil of 100 turns and dimensions 10.0 cm × 5.00 cm that rotates in a uniform magnetic field of 125 mT. The coil rotates at a steady 2340 rpm. (a) Find the maximum emf induced in the coil as it turns at this rate. (b) If the generator is connected to a 3.0 kΩ resistor what is the maximum power output (ignore resistance of the coil)? (c) Using the same resistor, what rotation rate would be necessary to double the power output?

18. A certain generator is connected to a 1.2 kΩ resistor. The current through the resistor peaks at 5.00 mA and reverses direction every 33 milliseconds. The coil inside the generator is a square loop of sides 2.0 cm and negligible resistance turning in a uniform magnetic field of 0.400 T. (a) What is the angular frequency of the rotating coil? (b) How many turns are on the coil? (c) How much does the coil rotate as the current drops from 5.00 mA to 2.50 mA and how much time does this take?

19. Suppose an electric drill is drilling a hole in a piece of wood. If the operator pushes harder on the drill, the electric motor slows down. (a) What is the effect on the current drawn by the electric motor? Explain. (b) What is the effect on the torque due to magnetic forces inside the motor? Explain.

20. A simple motor is powered by a 5.00 V source and consists of a coil rotating in a uniform magnetic field. It is observed that the current in the motor decreases from 165 mA to 45.0 mA as the rotation rate goes from zero to a steady 125 rpm. Determine the back emf as the motor turns at 125 rpm.

21. Imagine a DC electric motor which is completely frictionless and operating in a vacuum. If connected to a power supply of a given voltage would the motor attain a constant angular velocity? Or would it undergo angular acceleration indefinitely? Explain.

22. Explain how Maxwell’s modification to Ampere’s Law (the addition of the “displacement current”) is necessary to explain the existence of electromagnetic waves in which electric and magnetic fields oscillate.

23. If Maxwell’s equations are said to summarize all electrical and magnetic phenomena what happened to Coulomb’s Law (the inverse square law governing attraction and repulsion of charges)? Why is it not a necessary part of Maxwell’s equations?

24. A capacitor of 4700 μF is connected to a 3.00 V battery. (a) What total amount of charge will be stored in the capacitor? The battery is then disconnected and replaced with a 1.2 kΩ resistor. (b) What is the initial current in the resistor? (c) What will be the total heat generated in the resistor as the capacitor completely discharges?

25. In a certain defibrillator a capacitor stores electric energy in amount 160 J. The capacitor then totally discharges 0.18 C as the shock is delivered to the patient. (a) Find the required capacitance. (b) Determine the initial current if the patient’s chest has resistance 65 Ω.

26. A typical rechargeable battery supplies a voltage of 1.50 V and has a capacity of 450 mAh. (a) What capacitance would be required to store the same amount of energy at 1.50 V? (b) Could such a capacitor take on the roll of a battery in powering an electrical device? Explain.

27. A capacitor is formed by placing two circular metal plates of radius 20.0 cm parallel to one another and separated by 0.500 mm of air. (a) Find the capacitance of this configuration. (b) If connected to a 5.50 V battery how much charge develops on each plate? (c) An electric field of 3.0 × 10⁶ V/m between the plates would lead to a spark – based on this, what is the maximum energy that could be stored?

28. Suppose it is desired to form a capacitance of 10.0 μF using the same basic configuration as in the previous problem. (a) What separation is required if the same plates are used? (b) What size plates are required if the same separation is used? (Are either of these practical?)

29. A certain capacitor has parallel plates, area 175 cm² each, that are separated by 0.30 cm. The capacitor is charged by a 12.0 V battery and then it is disconnected. (a) Determine the amount of charge on each plate. (b) Find the work necessary to increase the separation of the plates to 0.50 cm (without discharging the plates).

30. Show that the plates of an “ideal” parallel plate capacitor attract each other with a force given by . Hint: determine the work necessary to increase the plate separation by an infinitesimal amount dx and then solve for force in the resulting expression.

31. A student makes a capacitor with a piece of paper sandwiched between two rectangular pieces of aluminum foil, 20.0 cm × 28.0 cm. The paper is 0.100 mm thick and has a dielectric constant 3.7 and a dielectric strength of 16000 kV/m. (a) Determine the capacitance. (b) What is the maximum voltage and energy storage for this capacitor?

32. A parallel plate capacitor has square plates, area 0.0350 m² each, separated by 1.20 mm. The capacitor is charged using a 12.0 V battery and then it is disconnected. An insulating slab of thickness 1.20 mm and dielectric constant, κ = 2.1 is then inserted between the plates. (a) Determine each of the following before and after the insertion of the slab: voltage, charge, electric field, potential energy. (d) Disregarding friction, what work must be done to insert the slab?

33. Repeat the previous problem but this time the capacitor remains connected to the battery during the insertion of the slab.

34. Determine a formula for the capacitance of a parallel plate capacitor with a dielectric slab that has a thickness exactly half that of the plate separation.

35. A spherical capacitor is formed by an interior metallic sphere of radius 5.00 cm surrounded by a metallic spherical shell of radius 5.10 cm. (a) Determine the capacitance. (b) Find the amount of energy stored if the capacitor is charged to 2.00 nC.

36. (a) Derive an expression for the capacitance of a cylindrical capacitor consisting of a metallic inner cylinder of radius R₁ and a metallic outer cylinder of radius R₂, each of the same length, L. Assume the length is much greater than the diameter. (b) If a capacitance of 2.0 pF is desired of a cylindrical capacitor of length 10.0 cm, what must be the ratio of the radii R₂ to R₁?

37. (a) Derive an expression for the capacitance of a single isolated charged sphere. (b) Use the result to determine the capacitance of a Van de Graff generator with a sphere of diameter 25.0 cm. (c) If the charge on this Van de Graff sphere reaches 3.0 μC, what is the voltage and energy associated with this “capacitor”?

38. Two capacitors C₁ = 30.0 μF and C₂ = 60.0 μF, are connected to a 12.0 V battery. Determine the voltage and amount of charge for each capacitor if the type of connection is: (a) series and (b) parallel. (c) Determine the effective total capacitance in each case.

39. Two capacitors are connected to a 6.00 V battery and a single-pole double-throw switch (SPDT) as illustrated below. The switch is in position 1 for a long while and then it is moved to position 2 and left there. Capacitor C₂ is initially uncharged. (a) Determine the eventual charge and voltage of each capacitor after the switch is in position 2. (b) Find the amount of charge that flows through the switch after it makes contact at position 2.

40. Examine the circuit below with capacitors: C₁ = 10.0 μF, C₂ = 25.0 μF, and C₃ = 40.0 μF. All the capacitors are initially uncharged as the switch is closed. After the switch has been closed for a long time, determine the following: (a) the voltage across C₁, (b) the charge on C₂, and (c) the energy stored in C₃.

41. A 6.0 V battery, a 220 W resistor, a 680 μF capacitor (initially uncharged), and a switch (initially open) are all connected in series, forming a loop. At t = 0 the switch is closed. (a) Find the initial current in the loop. (b) Find the charge on the capacitor at t = 0.100 s. (c) At what point in time will the voltage across the resistor equal 5.0 V? (d) What will be the maximum energy stored in the capacitor if the switch remains closed?

42. In the circuit described above the battery is removed after several minutes have passed. Then the loop is completed without the battery so that the capacitor is connected to the resistor. (a) How much time is required to discharge the capacitor by 99% (i.e. it retains only 1% of its original charge)? (b) What is the current at that point?

43. A student connects a charged capacitor to a resistor and measures the voltage across the resistor as the capacitor discharges. She finds that the voltage drops from 6.00 V to 1.00 V in 4.50 s. (a) How much additional time will be required for the voltage to drop from 1.00 V to 0.500 V? (b) If the resistor is 1.2 kW, what is the capacitance of the capacitor?

44. The capacitor is initially uncharged as the switch is moved to position 1 in the diagram below. (a) Find the initial current through the battery. (b) Find the eventual current through the battery after the switch has been in position 1 for a long time. (c) What is the charge on the capacitor at that point? (c) If the switch is moved to position 2, how much time will it take for the capacitor’s charge to drop by half?

45. Examine the circuit below in which the two capacitors are initially uncharged. (a) Find the initial current through the battery at the instant the switch is closed. (b) After the switch has been closed for a long time what is the charge on each capacitor? (c) What will be the initial current if the switch is then opened? (d) Make a careful sketch of current vs. time for the 22 W resistor illustrating the described actions.