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1 Chapter 33 Homework Due: 8:00am on Wednesday, April 7, 2010 Note: To understand how points are awarded, read your instructor's Grading Policy [Return to Standard Assignment View] Canceling a Magnetic Field Four very long, currentcarrying wires in the same plane intersect to form a square with side lengths of 460 through the wires are 80, 200, 100, and, as shown in the figure The currents running Find the magnitude of the current that will make the magnetic field at the center of the square equal to zero Hint A1 How to approach the problem Find the magnetic field at the center of the square due to the wires whose current you know Then, find the current wires whose contribution to the magnetic field will exactly cancel the contribution of the other three Hint A2 Calculating the contribution from the three known wires What is the magnitude of the magnetic field at the center of the square due to the wires carrying the 80, 20, and 10 currents? Be careful with signs when you add the contributions from the three different wires Hint A21 Ampère's law Recall Ampère's law: You can use this to determine the formula for the magnetic field generated by a long wire Use a circle centered on the wire as your path of integration Hint A22 Getting your signs correct Recall the righthand rule: If your thumb, on your right hand, points in the direction in which the current is flowing, your fingers will curl in the direction of the magnetic field Express your answer in teslas to three significant figures = Answer Requested Good You should have derived the following equation for the contribution to the magnetic field from one wire:, where is width of the square Use this formula and the fact that you want the magnetic field at the center to sum to zero to find the current Express your answer in amperes = 200 What is the direction of the current? Hint B1 How to approach the problem upward downward Magnetic Field due to Semicircular Wires A loop of wire is in the shape of two concentric semicircles as shown The inner circle has radius ; the outer circle has radius A current flows clockwise through the outer wire and counterclockwise through the inner wire 1 of 7 4/13/2010 8:05 PM
2 2 of 7 4/13/2010 8:05 PM What is the magnitude, Hint A1 Hint A2 Hint A3 Hint A4, of the magnetic field at the center of the semicircles? What physical principle to use Compute the field due to the inner semicircle Direction of the field due to the inner semicircle Compute the field due to the straight wire segments Express in terms of any or all of the following:,,, and = To see whether and makes sense, think of the scaling of different quantities The size of the current element scales as the radius, whereas the power of in the denominator is 2 (and equals the radius also, in this case) So over all, you would expect the magnetic field to scale as 1/radius Note that such an argument works only because the field due to each point is in the same direction, so you are doing a much simpler integral What is the direction of the magnetic field at the center of the semicircles? into the screen out of the screen The magnetic field at the center of a 0600cmdiameter loop is 250 Problem 3311 What is the current in the loop? 119 A A long straight wire carries the same current you found in part a At what distance from the wire is the magnetic field 250? m A 990 current circulates around a 160mmdiameter superconducting ring Problem 3316 What is the ring's magnetic dipole moment? What is the onaxis magnetic field strength 540 from the ring? T
3 3 of 7 4/13/2010 8:05 PM Problem 3340 A small bar magnet experiences a torque when the axis of the magnet is at to a 0150 magnetic field What is the magnitude of its magnetic dipole moment? 0198 Problem 3325 Magnetic resonance imaging needs a magnetic field strength of 15 T The solenoid is 18 m long and 75 cm in diameter It is tightly wound with a single layer of 180mmdiameter superconducting wire What current is needed? 2150 A Magnetic Force on Charged Particles Conceptual Question For each of the situations below, a charged particle enters a region of uniform magnetic field Determine the direction of the force on each charge due to the magnetic field Determine the direction of the force on the charge due to the magnetic field Hint A1 Determining the direction of a magnetic force points into the page points out of the page points neither into nor out of the page and Determine the direction of the force on the charge due to the magnetic field Hint B1 Determining the direction of a magnetic force points out of the page points into the page points neither into nor out of the page and
4 4 of 7 4/13/2010 8:05 PM Part C Determine the direction of the force on the charge due to the magnetic field Note that the charge is negative Hint C1 Effect of a magnetic field on a negative charge points out of the page points into the page points neither into nor out of the page and Problem 3326 A proton moves in the magnetic field with a speed of in the directions shown in the figure For each, what is magnetic force on the proton? Express vector in the form,,, where the x, y, and z components are separated by commas = 0, ,0 N Express vector in the form,,, where the x, y, and z components are separated by commas = 0,0,0 N The aurora is caused when electrons and protons, moving in the earth's magnetic field of Problem 3330, collide with molecules of the atmosphere and cause them to glow What is the radius of the cyclotron orbit for an electron with speed? 0114 m What is the radius of the cyclotron orbit for a proton with speed 104 m
5 5 of 7 4/13/2010 8:05 PM Magnetic Force on a CurrentCarrying Wire Learning Goal: To understand the magnetic force on a straight currentcarrying wire in a uniform magnetic field Magnetic fields exert forces on moving charged particles, whether those charges are moving independently or are confined to a currentcarrying wire The magnetic force on an individual moving charged particle depends on its velocity and charge In the case of a currentcarrying wire, many charged particles are simultaneously in motion, so the magnetic force depends on the total current and the length of the wire The size of the magnetic force on a straight wire of length carrying current in a uniform magnetic field with strength is Here is the angle between the direction of the current (along the wire) and the direction of the magnetic field Hence refers to the component of the magnetic field that is perpendicular to the wire, Thus this equation can also be written as The direction of the magnetic force on the wire can be described using a "righthand rule" This will be discussed after Consider a wire of length = 030 that runs northsouth on a horizontal surface There is a current of = 050 flowing north in the wire (The rest of the circuit, which actually delivers this current, is not shown) The Earth's magnetic field at this location has a magnitude of 050 (or, in SI units, ) and points north and 38 degrees down from the horizontal, toward the ground What is the size of the magnetic force on the wire due to the Earth's magnetic field? In considering the agreement of units, recall that Express your answer in newtons to two significant figures Because the Earth's magnetic field is quite modest, this force is so small that it might be hard to detect Now assume that a strong, uniform magnetic field of size 055 magnetic field pointing straight down is applied What is the size of the magnetic force on the wire due to this applied magnetic field? Ignore the effect of the Earth's Hint B1 Determining the angle theta Express your answer in newtons to two significant figures This force would be noticeable if the wire were of light weight The direction of the magnetic force is perpendicular to both the direction of the current flow and the direction of the magnetic field Here is a "righthand rule" to help you determine the direction of the magnetic force 1 Straighten the fingers of your right hand and point them in the direction of the current 2 Rotate your arm until you can bend your fingers to point in the direction of the magnetic field 3 Your thumb now points in the direction of the magnetic force acting on the wire Part C What is the direction of the magnetic force acting on the wire in due to the applied magnetic field? due north due south due east due west straight up straight down Part D
6 6 of 7 4/13/2010 8:05 PM Which of the following situations would result in a magnetic force on the wire that points due north? Check all that apply Current in the wire flows straight down; the magnetic field points due west Current in the wire flows straight up; the magnetic field points due east Current in the wire flows due east; the magnetic field points straight down Current in the wire flows due west and slightly up; the magnetic field points due east Current in the wire flows due west and slightly down; the magnetic field points straight down As you can see, many current/magnetic field configurations can result in the same direction of magnetic force Part E Assume that the applied magnetic field of size 055 is rotated so that it points horizontally due south What is the size of the magnetic force on the wire due to the applied magnetic field now? Hint E1 Determining the angle theta Recall that is the angle between the magnetic field direction and the direction of the current flow in the wire In this situation, what is the value of? 0 degrees 38 degrees 90 degrees 180 degrees Express your answer in newtons to two significant figures 0 Notice that whenever the current in the wire and the magnetic field point in the same direction ( ) or in opposite directions ( ), the sine of is zero, so there is no magnetic force exerted on the wire This is consistent with the earlier statement that it is the component of the magnetic field that is perpendicular to the direction of the current that produces the magnetic force Also notice that for these two special values of (when the current is flowing parallel to or antiparallel to the magnetic field) the steps listed for the righthand rule suggest a unique direction for the magnetic force This is another clue that the magnetic force is zero Problem 3336 What is the force on the first wire in the figure? What is the force on the second wire in the figure?
7 7 of 7 4/13/2010 8:05 PM Part C What is the force on the third wire in the figure? Rail Gun A Rail Gun uses electromagnetic forces to accelerate a projectile to very high velocities The basic mechanism of acceleration is relatively simple and can be illustrated in the following example A metal rod of mass and electrical resistance rests on parallel horizontal rails (that have negligible electric resistance), which are a distance apart The rails are also connected to a voltage source, so a current loop is formed The rod begins to move if the externally applied vertical magnetic field in which the rod is located reaches the value flattened bottom so that it slides instead of rolling Use for the magnitude of the acceleration due to gravity Assume that the rod has a slightly Find, the coefficient of static friction between the rod and the rails Hint A1 Hint A2 Hint A3 How to approach this problem Force due to the magnetic field Frictional force Express the coefficient of static friction in terms of variables given in the introduction = Score Summary: Your score on this assignment is 992% You received 7943 out of a possible total of 85 points, plus 485 points of extra credit
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