Review 0 is the magnetic permeability of free space whose value is Tm 0 4 10 7 A Physics for Scientists Engineers 2 The magnitude of the magnetic field at a distance r from a long straight wire carrying currrent i is given by B r Spring Semester 2005 Lecture 23 February 24 2005 Physics for Scientists Engineers 2 0i 2 r The magnitude of the magnetic field at the center of a loop with radius R carrying current i is given by i B 0 2R 1 Review 2 February 24 2005 Physics for Scientists Engineers 2 Force on a Current Carrying Wire Ampere s Law is B ds 0ienc Consider a long straight wire carrying carrying a current i in a constant magnetic field B where the integral is carried out around an Amperian loop and ienc is the current enclosed by the loop The magnetic field will exert a force on the moving charges in the wire The magnitude of the magnetic field inside a long wire with radius R carrying a current i at a radius r is given by The charge q flowing in the wire in a given time t in a length L of wire is given by L q ti i v i B r 0 2 r 2 R February 24 2005 2 where v is the drift velocity of the electrons Physics for Scientists Engineers 2 3 February 24 2005 Physics for Scientists Engineers 2 4 1 Force on a Current Carrying Wire 2 Parallel Current Carrying Wires The magnitude of the magnetic force is then Consider the case in which two parallel wires are carrying current L F qvBsin i vB iLBsin v The two wires will exert a magnetic force on each other because the magnetic field of one wire will exert a force on the moving charges in the second wire is the angle between the current and the magnetic field The direction of the force is perpendicular to both the current and the magnetic field and is given by the right hand rule The magnitude of the magnetic field created by a current carrying wire is given by This equation can be expressed as a vector cross product F iL B B r This magnetic field is always perpendicular to the wire with a direction given by the right hand rule iL represents the current in a length L of wire February 24 2005 Physics for Scientists Engineers 2 5 Parallel Current Carrying Wires 2 q2 ti2 0i1 2 d 6 L i2 v where v is the drift speed of the charge carriers Now consider wire two carrying a current i2 in the same direction as i1 placed a distance d from wire one The magnetic force is then L F qvB i2 vB1 i2 LB1 v The magnetic field due to wire one will exert a magnetic force on the moving charges in the current flowing in wire two Physics for Scientists Engineers 2 Physics for Scientists Engineers 2 The charge q2 flowing in wire two in a given time t in a length L of wire is given by The magnitude of the magnetic field a distance d from wire one is February 24 2005 February 24 2005 Parallel Current Carrying Wires 3 Let s start with wire one carrying a current i1 to the right B1 0i 2 r Putting in our expression for B 1 we get i ii L F12 i2 L 0 1 0 1 2 2 d 2 d 7 February 24 2005 Physics for Scientists Engineers 2 8 2 Torque on a Current Carrying Loop Torque on a Current Carrying Loop 2 Electric motors rely on the magnetic force exerted on a current carrying wire This force is used to create a torque that turns a shaft As the coil turns in the field the forces on the sides of the loop perpendicular to the magnetic field will change A simple electric motor is depicted below consisting of a single loop carrying current i in a constant magnetic field B The forces on the square loop with sides are illustrated below where is the angle between a normal vector n and the magnetic field B The two magnetic forces F and F shown in the figure are of equal magnitude and opposite direction These forces create a torque that tends to rotate the loop around its axis The normal vector is perpendicular to the plane of the wire loop and points in a direction given by the right hand rule based on the current flowing in the loop February 24 2005 Physics for Scientists Engineers 2 9 February 24 2005 Torque on a Current Carrying Loop 3 If we replace this loop with N loops wound close together we can write N 1 NiABsin Although we derived this expression for a square loop this express applies to circular loops as well as long as the magnetic field is uniform The force each of the vertical segments is F iaB The force on the other two sides is parallel or anti parallel to the axis of rotation and cannot cause a torque We can describe this coil with one parameter consisting of information about the coil only combined with information about the magnetic field The sum of the torque on the upper side plus the torque on the lower side gives the torque exerted on the coil about the center of the loop a a 1 iaB sin iaB sin ia 2 Bsin iABsin 2 2 We define the magnitude of the magnetic dipole moment of the coil above to be NiA where A a2 Physics for Scientists Engineers 2 10 Magnetic Dipole Moment Here the current is flowing upward in the top segment and downward in the lower segment as illustrated by the arrow feathers and arrowhead February 24 2005 Physics for Scientists Engineers 2 11 February 24 2005 Physics for Scientists Engineers 2 12 3 Magnetic Dipole Moment 2 The direction of the magnetic dipole moment is given by the right hand rule and points in the direction of the surface normal vector n Potential Energy of a Magnetic Dipole n A magnetic dipole has a potential energy in an external magnetic field If the magnetic dipole is aligned with the magnetic field it is in its minimum energy condition i If the magnetic dipole oriented in a direction opposite to the external field the dipole is in its maximum energy condition We can rewrite our expression for the torque as The magnetic potential energy U of a magnetic dipole in an external magnetic field B can be written as U B B cos NiA Bsin Bsin which we can generalize to B where is the angle between the magnetic dipole moment and the external field The torque will always be perpendicular the magnetic field magnetic dipole moment and the magnetic field February 24 2005 Physics for Scientists Engineers 2 This potential energy of orientation can be applied to many physical situations concerning magnetic dipoles in external magnetic fields 13 Magnetic Fields of Solenoids February 24 2005 Physics for Scientists Engineers 2 14 Magnetic Fields of …
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