Physics 25 Chapter 21 Dr. Alward Magnetism and Magnetic Forces Magnetic Field of a Bar Magnet Magnetic field lines flow away from the north pole and sink on the south pole. Like Poles Repel Unlike Poles Attract Magnetic Dipoles Magnets have two poles, one north, the other south. Bar magnets are magnetic dipoles. Magnetic dipole fields look just like the electric field of electric dipoles. A compass needle is a miniature bar magnet, one end of which (the red end) is north, and the other end of which is south. The north pole of the compass needle is attracted to Earth s south magnetic pole, which is near Prince Edward Island, Canada. Earth has a magnetic field that may be imagined to have been created by an 8,000 mile-long bar magnet, as suggested by the figures at the right. A compass needle lines up parallel to magnetic field lines. The red end of the compass barmagnet needle is its north pole, and points to Earth s south magnetic pole, which is the north geograhic pole. 1
Earth s Magnetic Field Intensity Earth s north magnetic pole is located at Earth s south geographic pole, and vice-versa. Magnetic field intensity is symbolized as B, and has units of tesla (T). Earth s magnetic field intensity near the ground is a very small fraction of a tesla, about ½ gauss. 1.0 T = 10,000 gauss (G) Magnetic Force on Moving Charges Use the right-hand rule to determine the direction of the force: Flatten palm, fingers, and thumb. Twist flat hand until the fingers point in the direction of the magnetic field arrows. Twist hand until the thumb points in the direction along which the positive charge is moving. The palm will then face in the direction of the force on the charge. If the charge is negative, the back of the hand faces in the direction of the magnetic force on the charge. To determine the value of the magnetic force, use the equation below: F = QvB sin θ θ is the angle between the thumb and index finger. Maximum force occurs when sin θ is maximum, at θ = 90 o. Charged particles moving parallel to magnetic field lines (θ = 0) experience zero magnetic force, because sin 0 = 0. Example A: Suppose a magnetic field created in a laboratory points from floor to ceiling, and an electron is fired in front of you, parallel to the ground, from your left to your right. In which direction will the electron be deflected? Answer: Away from you. Example B: A proton is fired upward from the equator. In which direction will it be deflected? Answer: westward Example C: A proton at the equator is fired northward. In which direction will it be deflected? Answer: Charges moving parallel to magnetic field lines experience zero magnetic force. 2
Example A: The magnetic field in a laboratory points from ceiling to floor. At Point P, a proton is fired from left to right, parallel to the floor. By the first right-hand rule, the magnetic force is perpendicular to the direction of motion, always pointing to the center of the circular path around which the proton travels. Derive an expression for the radius of the circular path around which the proton travels. F = ma evb = mv 2 /r eb = mv/r v =ebr/m r = mv/eb Example B: An electron and a proton, each having the same kinetic energy, enter a magnetic field. What is the ratio of the orbital radii? Note: the proton s mass is about 2000 times the electron s mass. First, note that K = ½ mv 2 = (mv) 2 /2m (mv) = (2mK) 1/2 ---------------------------------- r1/r2 = (m1v1)/eb / (m2v2) /eb = (2m1K1) 1/2 / (2m2K2) 1/2 = (m1/ m2) 1/2 = 2000 1/2 = 44.7 (proton radius / electron radius) 3
Magnetic Field Intensity due to a Long Straight Wire A current carrying straight wire is said to be long if its length is much greater than the distances of interest from the wire. For example, if you re one inch away from a ten-foot wire, or you re ten feet away from a 100-yard wire, then the wire can be regarded as long. The magnetic field consists of an infinite number of concentric circles whose planes are perpendicular to the wire, and which are of increasing greater and greater radii. Magnetic Constant: μo = 4π x 10-7 T-m/A B = μoi/2πr The arrow shows the direction along which imaginary positive charge carriers are moving, which is a direction opposite to the direction of flow of the actual charge carriers--the negatively charged electrons. This imagined current is called the conventional current. The shape of the magnetic field is determined by applying the second righthand rule : Hold the wire in your right hand with your thumb pointing in the direction of the conventional current. The circular direction of your fingers tells you the direction of the direction and shape of the magnetic field. The magnetic field due to the current in the wire is a set of concentric circles of progressively increasing radii. As viewed by one looking downward along the thumb axis, the magnetic field curves are counter-clockwise circles. Example: How close to a long straight wire carrying 0.5 A is the magnetic field intensity the same as Earth s B near the ground? Recall: Earth s magnetic field intensity near the ground is ½ gauss. 0.5 G = 5 x 10-5 T B = μoi/2πr 4π x 10-7 (0.5)/2πr = 5 x 10-5 r = 0.0020 m = 2.0 mm Example B: The magnetic field intensity at a certain distance from a long, straight, current-carrying wire is 270 G. What is the value of B one-third as far from the wire? Answer: Three times larger. B = 810 G 4
Example: At the right are shown two long parallel straight wires, separated by a distance 2d, and each carrying current I in the same direction. (a) What is B midway between the wires, at Point P? (b) What is B at Point Q, a distance d from Wire 2? (a) B1 = μoi/2πd (into sheet) B2 = μoi/2πd (out of sheet) Total = 0 (b) B1 = μoi/2π(3d) (into sheet) B2 = μoi/2πd (into sheet) Total = (μoi/2πd) (1/3 +1) = 2μoI/3πd Current-Carrying Loops: Equivalent to a Bar Magnet The second right hand rule shows that the B lines inside the loop point from right to left. This magnetic field is identical to a bar magnet s. (See below.) Apply the second right hand rule to see that the B arrows inside the loop-- where B is much larger than outside the loop-- point toward the viewer in the above figure, and away from the viewer in the figure at the right. If you re looking at a loop in which the current is counter-clockwise, you're looking at the "north" face of the equivalent bar magnet. On the left, the viewer sees the north pole of the equivalent bar magnet, while the south pole is seen above. If current is clockwise, you re looking at the south face of the magnet 5
B at the Center of a Circular Loop While a simple equation exists that allows a calculation of the magnetic field intensity anywhere for a long straight current-carrying wire, such is not the case for a circular current-carrying loop. For circular loops, we may easily calculated the value of B at the center of the loop, using the equation below: B = μoi/2r The example problem at the right puts both equations to use. Example: A circular loop of radius r = 4.0 cm is twisted into a long straight wire carrying current I = 2.5 A. What is the magnetic field intensity at the center of the loop, in gauss? B at the center is the sum of two contributions: one due to the straight wire, and one more from the loop. Both contributions point into the sheet, and therefore add: B = μoi/2πr + μoi/2r = 5.18 x 10-5 T = 51.8 x 10-6 T = 0.518 x 10-4 T = 0.518 G Magnetic Force on a Current-Carrying Wire The equation for the magnetic force on a straight current-carrying wire of length L carrying current I in a magnetic field whose intensity is B, is given by the equation below: Example: F = ILB sin θ θ is the angle between the directions of the conventional current and the magnetic field direction. In the figure at the right, θ = 90 o. What is the expression for the magnetic force F1 on the upper wire in the figure above? The magnetic field created by the lower wire--the field that is seen by the upper wire--is B2 = μoi2/2πr F1 = I1LB2 sin 90 = I1L(μoI2/2πr)(1) = μoi1i2l/2πr Note: the magnetic field created by Wire 1 plays no role in this discussion. 6