# Eðlisfræði 2, vor 2007

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2 = Part E The field found in this problem for a moving charge is the same as the field from a current element of length carrying current provided that the quantity is replaced by which quantity? Hint E.1 Making a correlation Force between Moving Charges Two point charges, with charges and, are each moving with speed toward the origin. At the instant shown is at position (0, ) and is at (, 0). (Note that the signs of the charges are not given because they are not needed to determine the magnitude of the forces between the charges.) What is the magnitude of the electric force between the two charges? Hint A.1 Which law to use Apply Coulomb's law: where is the distance between the two charges..2 Find the value of What is the value of for the given situation? Express your answer in terms of. = Express in terms of,,, and. = What is the magnitude of the magnetic force on due to the magnetic field caused by? Hint B.1 How to approach the problem First, find the magnetic field generated by charge at the position of charge. Then evaluate the magnetic force on due to the field of..2 Magnitude of the magnetic field The Biot-Savart law, which gives the magnetic field produced by a moving charge, can be written, where is the permeability of free space and is the vector from the charge to the point where the magnetic field is produced. Note we have in the numerator, not, necessitating an extra power of in the denominator. Using this equation find the expression for the magnitude of the magnetic field experienced by charge due to charge..2.a Determine the cross product What is the magnitude of? For any two vectors, Page 2 of 16

3 , where is the angle between the vectors. Because, in this case, is 45 degrees, Substitute the appropriate value of for this problem, to arrive at a surprisingly simple answer. Express your answer in terms of and. = Note that is cubed in the denominator of the Biot-Savart law. Express the magnitude of the magnetic field of (at the location of ) in terms of,,, and. =.3 Find the direction of the magnetic field Which of the following best describes the direction of the magnetic field from at? Remember, according to the Biot-Savart law, the field must be perpendicular to both and. Ignore the effects of the sign of. (along the x axis) (along the y axis) (along the z axis into or out of the screen) Hint B.4 Computing the force You can evaluate the force exerted on a moving charge by a magnetic field using the Lorentz force law:, where is the force on the moving charge, is the magnetic field, is the charge of the moving charge, and is the velocity of the charge. Note that, as long as and are perpendicular,. Express the magnitude of the magnetic force in terms of,,,, and. = Part C Assuming that the charges are moving nonrelativistically ( ), what can you say about the relationship between the magnitudes of the magnetic and electrostatic forces? Hint C.1 How to approach the problem Determine which force has a greater magnitude by finding the ratio of the electric force to the magnetic force and then applying the approximation. Recall that. The magnitude of the magnetic force is greater than the magnitude of the electric force. The magnitude of the electric force is greater than the magnitude of the magnetic force. Both forces have the same magnitude. This result holds quite generally: Magnetic forces between moving charges are much smaller than electric forces as long as the speeds of the charges are nonrelativistic. Biot-Savart Law: B-Field from Current Elements Magnetic Field from Current Segments Learning Goal: To apply the Biot-Savart law to find the magnetic field produced on the z axis from current elements in the xy plane. In this problem you are to find the magnetic field component along the z axis that results from various current elements in the xy plane (i.e., at ). The field at a point due to a current-carrying wire is given by the Biot-Savart law,, where and, and the integral is done over the current-carrying wire. Evaluating the vector integral will typically involve the following steps: Choose a convenient coordinate system--typically rectangular, say with coordinate axes,, and. Write in terms of the coordinate variables and directions (,, etc.). To do this, you must find and. Again, finding the cross product can be done either geometrically (by finding the direction of the cross product vector first, then checking for cancellations from any other portion of the wire, and then finding the magnitude or relevant Page 3 of 16

4 component) or algebraically (by using, etc.). Evaluate the integral for the component(s) of interest. In this problem, you will focus on the second of these steps and find the integrand for several different current elements. You may use either of the two methods suggested for doing this. The field at the point shown in the figure due to a single current element is given by, where and. In this expression, what is the variable in terms of and/or? Hint A.1 Making sense of subscripts Find, the z component of the magnetic field at the point from the current flowing over a short distance located at the point. Hint B.1 Cross product The key here is the cross product in the Biot-Savart law. What is the cross product when and are parallel? Express your answer in terms of,,,, and. Recall that a component is a scalar; do not enter any unit vectors. = 0 Part C Find, the z component of the magnetic field at the point from the current flowing over a short distance located at the point. Part C.1 What is Determine the displacement from the current element, the distance (magnitude) from the current element to the point in question? = Part C.2 Find the direction from the cross product Page 4 of 16

5 What is the unit vector that describes the direction of the magnetic field at the origin? Express your answer in terms of,, or. Express your answer in terms of,,,, and. Recall that a component is a scalar; do not enter any unit vectors. = Part D Find, the z component of the magnetic field at the point from the current flowing over a short distance located at the point. Part D.1 Determine the displacement from the current element Part D.2 Find the direction of the magnetic field vector What is the unit vector that describes the direction of the magnetic field at the origin? Hint D.2.a Evaluating cross products Express your answer in terms of,, or. Express your answer in terms of,,,, and. Recall that a component is a scalar; do not enter any unit vectors. = Part E Find, the z component of the magnetic field at the point P located at from the current flowing over a short distance located at the point. Part E.1 Part E.2 Determine the displacement from the current element Use the cross product to get the direction Express your answer in terms of,,,,, and. Recall that a component is a scalar; do not enter any unit vectors. = 0 Page 5 of 16

6 Part F Find, the z component of the magnetic field at the point P located at from the current flowing over a short distance located at the point. Part F.1 What is Determine the displacement from the current element, the displacement (magnitude) from the current element to the point in question? The figure shows another perspective of the same situation to make this calculation easier. = Part F.2 Determine which unit vector to use Another way to write the Biot-Savart law is, where you replace with. This eliminates the problem of finding and can make computation easier. You are asked for the z component of the magnetic field. points in the direction. Which component (,, or ) must you cross with to get a vector in the z direction? Recall that. Express your answer in terms of,, or (ignoring the sign). Part F.3 Evaluate the cross product The z component of the magnetic field results from the cross product of and the x component of. These two vectors are orthogonal, so finding the cross product is relatively straightforward. What is the value of? Give your answer in terms of,,, and. = Substitute this expression into the formula for the magnetic field given in the last hint. Observe that it has in the denominator since in the original equation was replaced with in the numerator. Express your answer in terms of,,,,, and. Recall that a component is a scalar; do not enter any unit vectors. = Magnetic Field at the Center of a Wire Loop A piece of wire is bent to form a circle with radius. It has a steady current flowing through it in a counterclockwise direction as seen from the top (looking in the negative direction). Page 6 of 16

7 What is, the z component of at the center (i.e., ) of the loop?.1.2 Specify the integrand Perform the integration Express your answer in terms of,, and constants like and. = 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 the inner wire. Magnetic Field due to Semicircular Wires flows clockwise through the outer wire and counterclockwise through What is the magnitude,, of the magnetic field at the center of the semicircles? Hint A 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. = Force between an Infinitely Long Wire and a Square Loop A square loop of wire with side length carries a current. The center of the loop is located a distance from an infinite wire carrying a current. The infinite wire and loop are in the same plane; two sides of the square loop are parallel to the wire and two are perpendicular as shown. Page 7 of 16

8 What is the magnitude,, of the net force on the loop? Hint A.1 How to approach the problem You need to find the total force as the sum of the forces on each straight segment of the wire loop. You'll save some work if you think ahead of time about which forces might cancel..2 Determine the direction of force Which of the following diagrams correctly indicates the direction of the force on each individual line segment? Hint A.2.a Direction of the magnetic field In the region of the loop, the magnetic field points into the plane of the paper (by the right-hand rule). Hint A.2.b Formula for the force on a current-carrying conductor The magnetic force on a straight wire segment of length, carrying a current with a uniform magnetic field along its length, is, where is a vector along the wire in the direction of the current. a b c d.3 Determine the magnitude of force Which of the following diagrams correctly indicates the relative magnitudes of the forces on the parallel wire segments?.3.a Find the magnetic field due to the wire What is the magnitude,, of the wire's magnetic field as a function of perpendicular distance from the wire,. Hint A.3.a.i Ampère's law Express the magnetic field magnitude in terms of,, and. = a b c d Page 8 of 16

9 .4 Find the force on the section of the loop closest to the wire What is the magnitude of the force on the section of the loop closest to the wire, that is, a distance from it? Hint A.4.a.4.b Formula for the force on a current-carrying conductor Find the magnetic field due to the wire Express your answer in terms of,,,, and. =.5 Find the magnetic field due to the wire What is the magnitude,, of the wire's magnetic field as a function of perpendicular distance from the wire,. Hint A.5.a Ampère's law Express the magnetic field magnitude in terms of,, and. = Express the force in terms of,,,, and. = The magnetic moment of a current loop is defined as the vector whose magnitude equals the area of the loop times the magnitude of the current flowing in it ( ), and whose direction is perpendicular to the plane in which the current flows. Find the magnitude,, of the force on the loop from in terms of the magnitude of its magnetic moment. Express in terms of,,,, and. = The direction of the net force would be reversed if the direction of the current in either the wire or the loop were reversed. The general result is that "like currents" (i.e., currents in the same direction) attract each other (or, more correctly, cause the wires to attract each other), whereas oppositely directed currents repel. Here, since the like currents were closer to each other than the unlike ones, the net force was attractive. The corresponding situation for an electric dipole is shown in the figure below. Ampere's Law and Examples Ampère s Law Explained Learning Goal: To understand Ampère s law and its application. Ampère s law is often written. The integral on the left is Part C Page 9 of 16

10 The circle on the integral means that must be integrated Part D Which of the following choices of path allow you to use Ampère s law to find? a. The path must pass through the point. b. The path must have enough symmetry so that is constant along large parts of it. c. The path must be a circle. Part E Ampère s law can be used to find the magnetic field around a straight current-carrying wire. Is this statement true or false? Part F Ampère s law can be used to find the magnetic field at the center of a square loop carrying a constant current. Is this statement true or false? Part G Ampère s law can be used to find the magnetic field at the center of a circle formed by a current-carrying conductor. Is this statement true or false? Part H Ampère s law can be used to find the magnetic field inside a toroid. (A toroid is a doughnut shape wound uniformly with many turns of wire.) Is this statement true or false? Magnetic Field inside a Very Long Solenoid Learning Goal: To apply Ampère's law to find the magnetic field inside an infinite solenoid. In this problem we will apply Ampère's law, written, to calculate the magnetic field inside a very long solenoid (only a relatively short segment of the solenoid is shown in the pictures). The solenoid has length, diameter, and turns per unit length with each carrying current. It is usual to assume that the component of the current along the z axis is negligible. (This may be assured by winding two layers of closely spaced wires that spiral in opposite directions.) From symmetry considerations it is possible to show that far from the ends of the solenoid, the magnetic field is axial. Which figure shows the loop that the must be used as the Ampèrean loop for finding for inside the solenoid? Page 10 of 16

11 .1 Choice of path for loop integral Part C Part D Part E Find, the z component of the magnetic field inside the solenoid where Ampère's law applies. Express your answer in terms of,,,, and physical constants such as. = Part F Part G The magnetic field inside a solenoid can be found exactly using Ampère's law only if the solenoid is infinitely long. Otherwise, the Biot-Savart law must be used to find an exact answer. In practice, the field can be determined with very little error by using Ampère's law, as long as certain conditions hold that make the field similar to that in an infinitely long solenoid. Which of the following conditions must hold to allow you to use Ampère's law to find a good approximation? a. Consider only locations where the distance from the ends is many times. b. Consider any location inside the solenoid, as long as is much larger than for the solenoid. c. Consider only locations along the axis of the solenoid. Hint G.1 Hint G.2 Implications of symmetry Off-axis field dependence Magnetic Field of a Current-Carrying Wire Find the magnetic field a distance from the center of a long wire that has radius and carries a uniform current per unit area in the positive z direction. First find the magnetic field,, outside the wire (i.e., when the distance is greater than ). Hint A.1.2 Ampère's law with current density Find the direction of the field Page 11 of 16

12 .3.4 Find the left-hand side of Ampère's law Find the right-hand side of Ampère's law Express in terms of the given parameters, the permeability constant, the variables, (the magnitude of ),,, and, and the corresponding unit vectors,, and. You may not need all these in your answer. = Now find the magnetic field inside the wire (i.e., when the distance is less than )..1.2 Establish the relationiship to Integrate over the Ampèrean loop Express in terms of the given parameters, the permeability constant, the distance from the center of the wire, and the unit vectors,, and. You may not need all these in your answer. = Magnetic Field inside a Toroid A toroid is a solenoid bent into the shape of a doughnut. It looks similar to a toy Slinky with ends joined to make a circle. Consider a toroid consisting of flowing through it. turns of a single wire with current Consider the toroid to be lying in the r plane of a cylindrical coordinate system, with the z axis along the axis of the toroid (pointing out of the screen). Let represent the angular position around the toroid, and let be the distance from the axis of the toroid. For now, treat the toroid as ideal; that is, ignore the component of the current in the direction. The magnetic field inside the toroid varies as a function of which parameters? Hint A.1 Consider rotational symmetry You were asked to assume that the toroid is rotationally symmetric, so its magnetic field cannot depend on. only only both and Inside the toroid, in which direction does the magnetic field point? (outward from the center) (inward toward the center) Page 12 of 16

13 (clockwise) (counterclockwise) (out of the screen) (into the screen) Notice that the direction is antiparallel to the path shown by the Ampèrean loop in the figure. Also, by definition,. Part C What is, the magnitude of the magnetic field inside the toroid and at a distance from the axis of the toroid? Hint C.1 Part C.2 Part C.3 How to approach the problem Evaluate the line integral Find Express the magnetic field in terms of, (the permeability of free space),, and. = Part D In an ideal toroid, current would flow only in the and directions. The magnetic field in the central plane, outside of the coils of such a toroid, is zero. For the toroid shown in the figures however, this field is not quite zero. This is because in this problem, there is a single wire that is wrapped around a doughnut shape. This wire must point somewhat in the direction, and thus the current must actually have a component in the direction. Compute, the magnitude of the magnetic field in the center of the toroid, that is, on the z axis in the plane of the toroid. Assume that the toroid has an overall radius of (the distance from the center of the toroid to the middle of the wire loops) and that is large compared to the diameter of the individual turns of the toroid coils. Note that whether the field points upward or downward depends on the direction of the current, that is, on whether the coil is wound clockwise or counterclockwise. Hint D.1 Simplifying the problem For this question, it is useful to consider the Biot-Savart law:. Since the question asks only for the z component of the magnetic field, we need only deal with those portions of that are parallel to (keeping in mind that ). In other words, the fact that the wire is wrapped in loops of radius is irrelevant; only the component of the current flow in the direction is important. With this simplification, the problem of finding the magnetic field at the center of the toroid becomes equivalent to finding the magnetic field at the center of a single circular loop of wire! Express in terms of,,,, and the local diameter of the coils. = This is the same expression that you would derive for the magnetic field at the center of a circular loop of current-carrying wire. To see why this makes sense, imagine that the local diameter the coils gets so small that it is negligible in comparison to the radius of the toroid. The wire makes one complete turn around the axis of the toroid. So, to a point in the center, the toroid looks like a simple current loop. of Current Sheet Consider an infinite sheet of parallel wires. The sheet lies in the xy plane. A current runs in the -y direction through each wire. There are wires per unit length in the x direction. Page 13 of 16

14 Write an expression for, the magnetic field a distance above the xy plane of the sheet. Use for the permeability of free space. Hint A Hint A.4.5 How to approach the problem Find Determine the direction of the magnetic field Magnitude of the magnetic field Evaluate Express the magnetic field as a vector in terms of any or all of the following:,,,,, and the unit vectors,, and/or. = Magnetic Fields in Matter Magnetic Materials You are given a material which produces no initial magnetic field when in free space. When it is placed in a region of uniform magnetic field, the material produces an additional internal magnetic field parallel to the original field. However, this induced magnetic field disappears when the external field is removed. What type of magnetism does this material exhibit? diamagnetism paramagnetism ferromagnetism When a paramagnetic material is placed in a magnetic field, the field helps align the magnetic moments of the atoms. This produces a magnetic field in the material that is parallel to the applied field. Once again, you are given an unknown material that initially generates no magnetic field. When this material is placed in a magnetic field, it produces a strong internal magnetic field, parallel to the external magnetic field. This field is found to remain even after the external magnetic field is removed. Your material is which of the following? diamagnetic paramagnetic ferromagnetic Very good! Materials that exhibit a magnetic field even after an external magnetic field is removed are called ferromagnetic materials. Iron and nickel are the most common ferromagnetic elements, but the strongest permanent magnets are made from alloys that contain rare earth elements as well. Part C What type of magnetism is characteristic of most materials? ferromagnetism paramagnetism diamagnetism no magnetism Almost all materials exhibit diamagnetism to some degree, even materials that also exhibit paramagnetism or ferromagnetism. This is because a magnetic moment can be induced in most common atoms when the atom is placed in a magnetic field. This induced magnetic moment is in a direction opposite to the external magnetic field. The addition of all of these weak magnetic moments gives the material a very weak magnetic field overall. This field disappears when the external magnetic field is removed. The effect of diamagnetism is often masked in paramagnetic or ferromagnetic materials, whose constituent atoms or molecules have permanent magnetic moments and a strong tendency to align in the same direction as the external magnetic field. Forces between a Charge and a Bar Magnet Learning Goal: To understand the forces between a bar magnet and 1. a stationary charge, 2. a moving charge, and 3. a ferromagnetic object. A bar magnet oriented along the y axis can rotate about an axis parallel to the z axis. Its north pole initially points along. Interaction of stationary charge and bar magnet A positive charge is displaced some distance in the direction from the magnet. Page 14 of 16

15 Assume that no charges are induced on the magnet. Assume that the length of the magnet is much smaller than the separation between it and the charge. As a result of magnetic interaction (i.e., ignore pure Coulomb forces) between the charge and the bar magnet, the magnet will experience which of the following? Interaction of moving charge and bar magnet Consider a second case in which the charge is again some distance in the direction from the magnet, but now it is moving toward the center of the bar magnet, that is, with its velocity along. Due to its motion in the magnetic field of the bar magnet, the charge will experience a force in which direction?.1 Determine the magnetic field direction near a charge.2 Determine the direction of force on a charge moving in a magnetic field Interaction of iron and bar magnet Now the charge is replaced by an electrically neutral piece of initially unmagnetized soft iron (for example, a nail) that is not moving. Part C As a result of the magnetic interaction between the soft iron and the bar magnet, which of the following will occur? Hint C.1 Magnetic induction Summary 5 of 13 problems complete (28.67% avg. score) 17.3 of 22 points Page 15 of 16

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