B 1 Magnetism a new force So far we ve learned about two forces gravity and the electric field force G G G G FE Definition of E field E FE q E q G kQ E fields are created by charges E 2 r G G E field exerts a force on other charges FE q E E The gravitational force is similar G GM Gravitational fields are created by mass g 2 r G G The gravitational field exerts a force on other masses Fgrav m g There is a different kind of field called a magnetic field or B field B fields are created by moving charges currents B fields exert forces on moving charges Remember I is to B as q is to E We will see in the next chapter exactly how B fields are made by currents For now we assume that we have a B field and we want to know how the B field exerts a force on a moving charge a current The force FB from a B field on a moving charge depends on the velocity of the charge in a G G peculiar way a charge q moving with velocity v in a magnetic field B feels a magnetic force K K K given by FB q v B q times cross product of v and B see appendix for review of the K K cross product This equation is the definition of B analogous to the FE q E which defines E G The magnitude of the magnetic force is FB FB q v B sin q v B B v v sin B v v q Last update 10 13 2009 Dubson Phys1120 Notes University of Colorado B 2 G G G The direction of the force FB is perpendicular to the plane formed by v and B The direction of K K v B is determined by the Right hand rule Use your right hand Point your fingers in the G G direction of v curl your fingers toward B Orient hand so that fingers curl thru angle 180o G K K Your thumb then points in the direction of v B which is the direction of FB if the charge q is G positive If q is FB is other way v B B v B q F out F in B v q v B F in v q Unlike gravity or the electric force the magnetic force is a velocity dependent force G G If v B then sin 1 FB q v B G G If v B then sin 0 FB 0 If v 0 then FB 0 Units of B B B unlike gravity or E field force F N 1 tesla T m q v C s Older non SI unit of B 1 gauss 10 4 T 1 T 104 gauss Earth s magnetic field 0 5 gauss 5 10 5 T kitchen magnet 50 500 gauss 0 005 0 05 T iron core electromagnet 2 T max Strong enough to yank tools out of your hand superconducting magnet 20 T max Last update 10 13 2009 Dubson Phys1120 Notes University of Colorado B 3 We have said that currents make B 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 moving charge forming a tiny current loop an atomic current In most metals the atomic currents of different atoms have random orientations so there is no net current no B field But 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 q moving in a uniform magnetic field B Since the force FB is always perpendicular to the velocity v the force FB does no work K G K K G K dr FB v v WFB FB dr 0 A magnetic force cannot change the KE of a dt particle recall Work KE theorem Wnet KE The B field changes the direction of the velocity v but does not change the speed so we have v constant If the velocity v is perpendicular to the field B the magnetic force bends the path of the particle in a circle q v FB q FB v FB v B out uniform B out R FB v We can relate the radius R of the circular path to the magnitude of the field B and the speed v with Newton s Second Law Last update 10 13 2009 Dubson Phys1120 Notes University of Colorado B 4 G G v2 Fnet m a q v B m R Solving for R we get R G v2 recall that for circular motion a R mv Notice that the radius is proportional to the mass of the qB particle In a mass spectrometer the mass of an unknown particle is determined from measurement of the radius assuming charge speed and B field are all known B Since the magnetic force has no component along the direction of B there is no acceleration in that direction and the component of the velocity along the direction of B is constant Consequently charged particles moving in a magnetic field can form spiral trajectories spiraling around and along the B field lines as shown Charged particles protons from the sun solar wind are guided along the earth s B field to arctic regions where they slam into the atmosphere producing Northern lights The Velocity Selector The velocity selector is a device which measures the speed v of an ion ion charged atom with one or more electrons missing A magnet produces a uniform Bfield and a capacitor produces a uniform E field with E B Forces on charge v E B in FB q v B FE q E mass m charge q uniform E down B in The B and E fields are adjusted until the particle goes straight through If the path is straight then FB FE qvB qE v E B Last update 10 13 2009 Dubson Phys1120 Notes University of Colorado B 5 Magnetic force on a current carrying wire A B field exerts a force on a moving charge A current carrying wire is full of moving charges so a B field exerts a force on the current carrying wire The force on a straight wire of length L K K K carrying a current I in a uniform magnetic field B is given by F I L B where we define a length vector L having magnitude L length of the wire and direction equal to the direction of the current in the wire Proof N moving charges K K K Force on a single charge Fon q q v B q area A Number of charges in wire N N n A L N vol B v L volume K K K K K Total force on all the charges Ftot N q v B n A L q v B Recall from the chapter on …
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