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Tutorial 1 Calculating the Magnetic Force on a Wire Sample Problem 1: Calculating the Magnitude of the Magnetic Force on a Wire in a Uniform Magnetic Field A piece of wire 45.2 cm long has a current of 12 A (Figure 5). Solution: For wire = /LB sing The wire moves through a uniform magnetic field with a strength of 0.30 T. Calculate the magnitude of the magnetic force on the = (125 )(0.452 m)(0.30 9 sing wire when the angle between the magnetic field and the wire is = (1.627 kg-m/s?) sine (a) 0, (b) 45%, and (c) 90". Fan wine = (1.627 N)sine (two extra digits carried) (a) When # = 0', then sin e = 0, so Fan wire = (1.627 N) (0) 1 = 12 A Fan wire = ON (b) When o = 45', then sin # = 0.707, so Fan wire = (1.627 N) (0.707) L - 45.2 cm Fan wire = 1.2 N Figure 5 (c) When # = 90", then sin o = 1, so Given: / = 12A; L = 45.2 cm = 0.452 m; B = 0.30 T Required: For wim Fanwire = (1.627 N)(1) Fan wing = 1.6 N Analysis: Use the equation for the magnitude of the magnetic force on a current-carrying wire: Fin = /LB sin 6. Note that Statement: The magnitude of the force on the wire is O N when 1 A = 1 C/s. e = 0', 1.2 N when e = 45', and 1.6 N when e = 90. Sample Problem 2: Determining Magnetic Force on a Segment of a Current-Carrying Wire in Earth's Magnetic Field Two electrical poles support a current-carrying wire. The mass Solution: of a 2.5 m segment of the wire is 0.44 kg. A 15 A current travels Fan wire = /LB sine through the wire. The conventional current is oriented due east, horizontal to Earth's surface. The strength of Earth's magnetic = (15)(2.5m) (5.7 x 10-5 ke (sin 90%) field at the location is 57 pT and is oriented due north, horizontal to Earth's surface (Figure 6) = 2.1 x 10-3 kg-m/s? Fan wine = 2.1 x 10 'N north Using the right-hand rule, point the fingers of your right hand in the direction of the current, east. Next, curl your fingers in - east the direction of the magnetic field, north. Your thumb points in the direction of the resulting magnetic force, upward. Statement: The magnitude of the magnetic force on the Figure 6 2.5 m segment of wire is 2.1 x 10 N. The force is (a) Determine the magnitude and the direction of the magnetic directed upward. force on the 2.5 m segment of wire. (b) Given: m = 0.44 kg (b) Calculate the gravitational force on the 2.5 m segment of wire. Required: F Solution Analysis: F. = mg (a) Given: 8 = 57 pT = 5.7 x 10 # T; / = 15A; L = 2.5 m; Solution: F. = mg # = 90 = (0.44 kg) (9.8 m/s?) Required: For wire F. = 4.3 N Analysis: Use the equation Fan zine = /LBsine to determine Statement: The gravitational force on the 2.5 m segment of the magnitude of the magnetic force; then use the right- wire is 4.3 N. hand rule for a current-carrying wire in a magnetic field to determine the direction.8.3 Magnetic Force on a Current-Carrying Conductor Other than the musicians, probably the most important parts of a concert are the speakers. You might have seen the large boxes that transmit sound to both the per- formers and the audience (Figure 1). Speakers are so common that we rarely think about the impact this invention has had on our culture. People use speakers at music concerts, in cars, and in portable devices. Telephones, computers, and televisions all rely on speakers to transmit sound. Intercom systems in buildings, megaphones used by firefighters and police officers, and even electronic readers for sight-impaired people are all possible because of the speaker. Figure 1 Musicians rely on the magnetic force in speakers to transmit sound during a performance. How does a speaker work? Inside a speaker is an electromagnet as well as a per- manent magnet. The magnetic field of the permanent magnet exerts a force on the current in the coil of the electromagnet. The speaker uses this force to produce sound waves. Simply put, a speaker works because a current-carrying conductor experi- ences a force in a magnetic field. Magnetic Force and Current An electric current consists of a collection of moving charges. We can therefore use the formula for the magnetic force on a single moving charge to determine the mag- netic force on a current-carrying wire. This force is important in many applications, including electric motors. Consider a current-carrying wire placed in an external magnetic field as shown in Figure 2. The magnetic field B is uniform and perpen- dicular to the wire. An external force, not the current in the wire, produces this field. We can calculate the magnetic force on the wire due to this external field by adding the magnetic forces on all the moving charges in the wire. B Figure 2 This current-carrying wire is in an external magnetic field. The magnetic field is perpendicular to the wire and is directed out of the page as indicated by the dots.A segment of the wire of length L is shown in Figure 3. The current, I, in this segment is At where q is the electric charge that passes by one end of the wire segment in a time interval At. g Figure 3 In our analysis to determine the magnetic force on a current-carrying wire, we examine a section of the wire and call it length L This equation is the relationship between charge and current you have studied before. The magnetic force on this moving charge is given by Fu = qvB sine The speed of the charge is just V = At Substituting the speed, v, in the magnetic force equation gives Fu = qvB sine L B sin 0 = 4 At FM = 4 LB sine Using the relationship between current and charge, I =- At' , we get FM = ILB sin e The magnetic force on the moving charge is really a force on the wire. We can rewrite the equation to express this relationship explicitly: Fan wire = /LB sine where o represents the angle between I and B. Figure 4 shows how to use the right-hand rule to determine the direction of Fom wing. Begin with the fingers of your right hand in the direction of the magnetic field and point your thumb in the direction of the current. Your palm then points in the direc- tion of the force on the wire. The external magnetic field pulls this wire downward as long as the current is directed to the right. The magnetic force on a current is due to the force on a moving charge, so the right-hand rule used here is similar to other right-hand rules you have used before. The direction of the current replaces the direc- tion of v for a moving charge. Figure 4 You can use the right-hand rule to determine the direction of the magnetic force on the wire. In the following Tutorial, you will calculate the magnetic force on a current- carrying wire in a magnetic field.Loudspeakers Figure 8 shows how a loudspeaker uses a magnetic force on a current to produce sound. The wire coil inside the speaker is part of an electromagnet. Electrical signals corresponding to sounds produce a changing current in the coil. The changing cur- rent produces a changing magnetic field around the coil. The permanent magnet also has a magnetic field. This field exerts a force on the current-carrying wire. Variations in the current produce variations in the force on the wires in the coil. The coil moves back and forth in response. The vibrating coil causes the cone to vibrate, pushing sound waves through the air and into your ears. magnetic material solenoid SN permanent magnet applied current speaker cone Figure 8 A loudspeaker changes electrical signals to sound using a permanent magnet and an electromagnet. Electromagnetic Pumps An understanding of the magnetic force on a current enabled medical researchers to devise electromagnetic pumps to move fluids in kidneys and artificial hearts. Traditional mechanical pumps can cause damage to blood cells. The use of magnetic fields eliminates this problem. Scientists are able to keep the blood flowing to the heart during kidney dialysis, for example, by creating a magnetic field over tubes of blood containing an electric current. A magnetic force acting on the charged particles keeps the blood in motion. WEB LINK