Physics E1bx: Assignment for Mar Mar. 31 Homework #5 Magnetic Fields Due Tuesday, Mar. 24, at 6:00 PM

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1 Physics E1bx: Assignment for Mar Mar. 31 Homework #5 Magnetic Fields Due Tuesday, Mar. 24, at 6:00 PM This assignment must be turned in by 6:00 pm on Tuesday, Mar. 31. Late homework can not be accepted. Please write your answers to these questions on a separatetype equation here. sheet of paper with your name and your section TF s name written at the top. Turn in your homework to the mailbox marked with your section TF s name in the row of mailboxes outside of Sci Ctr 108. If you submit your homework online, it must be in a.pdf format. You are encouraged to work with your classmates on these assignments, but please write the names of all your study group members on your homework. After completing this homework assignment, you should be able to Know the similarities and differences between magnetic and electric field lines Use the right-hand rule to find the force on a moving charge Find the magnitude and direction of the magnetic field produced by a current-carrying wire Find the magnetic force on a wire Calculate magnetic flux Apply Faraday s law to calculate the induced EMF Apply Lenz s law to determine the direction of the induced current Find the magnetic energy of a loop (dipole) in an external magnetic field 1

2 Physics E1bx Mar. 24, 2015-Mar. 31, 2015 Here are summaries of this module s important concepts to help you complete this homework: 2

3 Physics E1bx Mar. 24, 2015-Mar. 31,

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5 1. Putting Everything Together (Exam-Type Question): Lenz s Law (3 pts) Five loops are formed of copper wire of the same gauge (cross-sectional area). Loops 1 4 are identical; loop 5 has the same height as the others but is longer. At the instant shown, all the loops are moving at the same speed in the direction indicated. There is a uniform magnetic field point into the page in region I; in region II there is no magnetic field. Ignore any interactions between the loops. a) For any loop that has an induced current, indicate the direction of the current See blue arrows 4 b) Rank the magnitudes of the emfs around the loops. Explain your reasoning. EMFs are proportional to the change in flux. 2 and 3 have no change in flux, thus no EMF. The EMFs of 1, 4, and 5 is all the same magnitude, which is non-zero. This is because Region I they are the same height and same speed so the amount of B-field going through them changes equally over time. c) Rank the magnitudes of the currents in the loops. Explain your reasoning. 5 Region II There is zero current in 2 and 3. There is a large current in 1 and 4, that is, it is large in comparison to the current in 5, which is less (but non-zero) because there is more wire, thus more resistance (recall resistivity resistance relation). 5

6 2. Putting It Together: The Mass Spectrometer (3 pts) Potassium ions (charge -e, mass m K = 39 m p ) are accelerated from rest at plate A through a voltage difference of ΔV = 5 kv to plate B. Some of the ions pass through a small slit in plate B to enter into a region of uniform magnetic field (the shaded region in the diagram). m p is the mass of a proton, kg. 6

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8 3. Magnetic dipole loop (4 pts) A square wire loop with side length L and current i, lying in the x-z plane, is placed in a uniform magnetic field of magnitude B pointing in the positive x-direction. (Assume the magnitude of the current in the loop doesn t change throughout this problem.) y i 4 B a) Write an expression for the magnitude and direction of the force on each side of the loop (the sides are labeled from 1 to 4). Around which axis would the loop rotate due to these forces? z x The force is zero on leg 1 and 3 because it is parallel to the B-field. That is the cross product in F = il B is zero. The magnitude of the force on sides 2 and 4 is ilb. This force is upward on side 4 and downward on side 2 (due to the right-hand rule). So, it would rotate counterclockwise around the z-axis. b) Write an expression for the magnetic energy of the loop in the magnetic field, in terms of quantities given in the question, and simplify as much as possible. (hint: treat the loop at a magnetic dipole). The magnetic dipole in a magnetic field has energy U mag =! µ! B where! µ = i! A. In this orientation the area vector is perpendicular to the magnetic field so the dot product is zero and the magnetic energy is zero. c) Imagine that the loop has rotated 90 degrees around the axis that you indicated in part a) (in the direction that the forces would cause the loop to rotate). What is the magnitude and direction of the force on each side of the loop and the net force on the loop, in this new orientation? The magnitude of the force on each side is ilb. The direction on side 1 is in the positive z direction; side 2 negative y direction; side 3 negative z direction; side 4 positive y direction. These forces sum to zero net force. d) What is the magnetic energy of the loop in this new orientation? Is the magnetic energy of the loop now bigger or smaller than when in its initial orientation? The magnetic dipole in a magnetic field has energy U mag =! µ! B where! µ = i! A. So here, that means -L 2 ib. This is smaller than the initial energy of zero. e) BONUS (interesting, but not required): How is this similar to a ball rolling down a hill into a valley Where does the ball have more potential energy? Where does the loop have more magnetic energy? Where does the ball have zero force on it? Where does the loop have zero force on it? Do forces seem to push objects to places of higher or lower potential energy? 8

9 The ball has a higher potential energy near the top of the hill, and the loop has a higher potential energy when it is in its initial orientation. When the ball is in a valley it has no net force on it, and when the loop is in its final position it has no force on it. It seems the force moves objects to lower potential energy. 9