Magnetism: a new force!


 Maud Cassandra Fisher
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1 1 Magnetism: a new force! So far, we'e learned about two forces: graity and the electric field force. =, = Definition of field kq fields are created by charges: = r field exerts a force on other charges: =. The graitational force is similar: GM Graitational fields are crea ted by mass: g =. r The graitational field exerts a force on other masses. = mg. gra There is a different kind of field, called a magnetic field or field. fields are created by moing charges (currents). fields exert forces on moing charges. Remember: is to as is to. We will see in the next chapter exactly how fields are made by currents. or now, we assume that we hae a field and we want to know how the field exerts a force on a moing charge (a current). The force from a field on a moing charge depends on the elocity of the charge in a peculiar way: a charge, moing with elocity in a magnetic field, feels a magnetic force gien by = ( times crossproduct of and : see appendix for reiew of the crossproduct). This euation is the definition of, analogous to the =, which defines. The magnitude of the magnetic force is = = sin = = sin Dubson Phys110 Notes, Uniersity of Colorado
2 The direction of the force is perpendicular to the plane formed by and. The direction of is determined by the "Righthand rule". Use your right hand. Point your fingers in the direction of, curl your fingers toward. (Orient hand so that fingers curl thru angle < 180 o ). Your thumb then points in the direction of, which is the direction of if the charge is positie. f is ( ), is other way  (out) (in) (in) ( ) Unlike graity or the electric force, the magnetic force is a elocitydependent force. f, then sin = 1 = f then sin = 0 f = 0, then = 0 (unlike graity or field force) Units of : [] = [] = [] N [][] = m = C s 1 tesla (T) Older, nons, unit of : 1 gauss = 104 T, 1 T = 10 4 gauss arth's magnetic field 0.5 gauss = T kitchen magnet : gauss = T iron core electromagnet: T (max) (Strong enough to yank tools out of your hand.) superconducting magnet: 0 T (max) Dubson Phys110 Notes, Uniersity of Colorado
3 3 We hae said that currents make fields. So where's the current in a permanent magnet (like a compass needle)? An atom consists of an electron orbiting the nucleus. The electron is a moing charge, forming a tiny current loop an "atomic current". n most metals, the atomic currents of different atoms hae random orientations, so there is no net current, no field. ut in magnetic materials the atomic currents are aligned and they create a net current. (More on permanent magnets in the next chapter.) Motion of a charged particle in magnetic field Consider a charge moing in a uniform magnetic field. Since the force is always perpendicular to the elocity, the force does no work : dr, = W = dr = 0. A magnetic force cannot change the K of a dt particle (recall WorkK theorem: W net = K). The field changes the direction of the elocity, but does not change the speed, so we hae = constant. f the elocity is perpendicular to the field, the magnetic force bends the path of the particle in a circle. uniform (out) (out) R We can relate the radius R of the circular path to the magnitude of the field and the speed with Newton's Second aw: Dubson Phys110 Notes, Uniersity of Colorado
4 4 net = m a = m R ( recall that for circular motion a = R ) Soling for R, we get R m =. Notice that the radius is proportional to the mass of the particle. n a massspectrometer, the mass of an unknown particle is determined from measurement of the radius (assuming charge, speed and field are all known). Since the magnetic force has no component along the direction of, there is no acceleration in that direction, and the component of the elocity along the direction of is constant. Conseuently, charged particles moing in a magnetic field can form spiral trajectories, spiraling around and along the field lines as shown. Charged particles (protons) from the sun (solar wind) are guided along the earth's field to arctic regions, where they slam into the atmosphere, producing "Northern lights". The Velocity Selector The elocity selector is a deice which measures the speed of an ion. (ion = charged atom with one or more electrons missing). A magnet produces a uniform  field and a capacitor produces a uniform field, with. mass m charge (in) uniform down, (in) orces on charge: = = The and fields are adjusted until the particle goes straight through. f the path is straight, then = = = /. Dubson Phys110 Notes, Uniersity of Colorado
5 5 Magnetic force on a currentcarrying wire A field exerts a force on a moing charge. A currentcarrying wire is full of moing charges, so a field exerts a force on the currentcarrying wire. The force on a straight wire of length, carrying a current, in a uniform magnetic field, is gien by =, where we define a length ector, haing magnitude = length of the wire and direction eual to the direction of the current in the wire. Proof: orce on a single charge = on =. Number of charges in wire = N = n A area A #/ ol olume Total force on all the charges = = N = na tot N moing charges Recall from the chapter on current that = na =. tot J = = n = na so A Alternatie proof: et t is time for charge to moe distance, so speed of moing charges = =, current t N =. Assume wire, just to simplify math. t N on wire = N on = N = =. Done. t t (uniform) (in) = ( direction) f wire is not straight or is not uniform, then do that calculus thing: in your imagination, break the wire up into little segments d. d = d, = d = d tot d d (in) Dubson Phys110 Notes, Uniersity of Colorado
6 6 orce on a Current oop New term: "magnetic dipole moment" or "magnetic moment" = a loop of current magnetic moment µ = current area µ µ = A current Direction of µ = direction of area ector A + righthand rule: area A curl fingers of right hand in direction of current, thumb points along µ. (t's called a magnetic dipole moment because the field created by the current loop looks similar to the field created by an electric dipole. More on that in next chapter.) Suppose loop is in a uniform field. Then net = 0, always, since forces on opposite sides of loop will cancel. (out) = A µ (in) ut if µ is not parallel to (loop not a perpendicular to ) then there is a torue τ, (out) tending to twist the loop so that µ aligns with. side iew of loop t's not hard to show that the torue is τ = A = µ. This euation τ = µ is the principle of operation of a galanometer, which is a deice that measure current: the greater the current, the greater the torue which causes a needle to rotate along a calibrated scale. lectric Motors An electric motor is a deice which conerts electrical energy into mechanical work. t consists of a rotating coil of wire carrying current in a constant magnetic field. (The field is made by a permanent magnet or by another coil of wire with current.) Dubson Phys110 Notes, Uniersity of Colorado
7 7 The field exerts forces on the coil, causing it to rotate. After the coil rotates (out) 180 o, the current reerses direction so that the force always causes the coil to rotate in the same sense. (uniform) rotation axis (in) So far, we hae assumed the existence of, and described the force on a moing charge due to that. Now we will show how to make a field with a current. CrossProduct Reiew: The crossproduct of two ectors is a third ector A = C defined like this: The magnitude of A is A sin. The direction of A is the direction perpendicular to the plane defined by the ectors A and plus righthand rule. (Curl fingers from first ector A to second ector, thumb points in direction of A A A Dubson Phys110 Notes, Uniersity of Colorado
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