Lorentz Force. Velocity Selector

Transcription

1 Lecture 9-1 Lorentz Force Let E and denote the electric and magnetic vector fields. The force F acting on a point charge q, moving with velocity v in the superimosed E fields is: F qe v This is called the Lorentz force equation. Velocity Selector E Assume that the electric field is created by a parallel plate capacitor pointing along the Y axis and the magnetic field along the Z axis as shown in the figure below. Since the moving charged particle is negative: F F E y y qe qv (up) (down)

2 Lecture 9-2 When the electric and magnetic forces balance qe qv 0 v E Y X

3 Lecture 9-3 Magnetic Mass spectrometer When an ion of unknown mass enters the homogeneous magnetic field of the Magnetic spectrometer it executes a circular path. The measured radius of the circle depends on its mass m and its velocity. Thus v also has to be measured. (See Lec.8, page 12) v

4 Lecture 9-4 Physics 219 Question 1 Sept A proton (charge +e) comes horizontally into a region of perpendicularly crossed, uniform E and fields as shown. In this region, it deflects upward as shown. What can you do to change the path so it remains horizontal through the region? a) Increase E b) Increase c) Turn off d) Turn E off e) Nothing e +

5 Lecture 9-5 Magnetic Force on a Current A Consider a straight current-carrying wire in the presence of a magnetic field. There will be a force on each of the charges moving in the wire. What will be the total force F on a length L of the wire? Current is made up of n charges/volume, each carrying charge q < 0 and moving with velocity v d through a wire of cross-section A. Force on each charge = Total force: qv F n( AL) qv On the next page we show that the product navq is equal to the current flowing in the wire. F il

6 Lecture 9-6 In time t, all the free charges in the shaded volume pass through A. If there are n charge carriers per unit volume, each with charge q, the total free charge in this volume is Q qnav is dt, where vd the drift velocity of the charge carriers.

7 Lecture 9-7 Magnetic Force on a Current Loop Force on closed loop current in uniform? Force on top path cancels force on bottom path (F = IL) Force on right path cancels force on left path. (F = IL) F I L loop I L loop closed loop 0 Uniform exerts no net force on closed current loop.

8 Lecture 9-8 Magnetic Torque on a Current Loop a b If field is to plane of loop, the net torque on loop is 0. definition of torque r F abut a chosen point If field is // to plane of loop, the net torque on loop is maximum. b n n magnetic moment direction so that n is twisted to align with

9 Lecture 9-9 Thus: Calculation of Torque Suppose the coil has width b (the side we see) and length a (into the screen). The torque about the center is given by: b r F 2 sin F 2 magnetic moment Iab sin (Simple Approach) The magnitude of magnetic force F acting on a wire of length a carrying current I in magnetic field is: F Ia IAsin area of loop Note: if loop, = 0 and = 0. Maximum torque occurs when plane of the loop is parallel to.

10 Lecture 9-10 Calculation of Torque (General Approach) For reference : Giambattista, Vol. 2, Ch. 19., page 719., prob. 47. It can be shown that the magnetic moment of planar loop of any shape of area A carrying a current I is: NIA N denotes the number of turns. The magnetic torque in magnetic field is: A NIA (19.13 b) The direction of is can be determined from the current direction in the loop and the right hand rule. The direction of is along the rotation axis. The direction can be obtained by the right hand rule.

11 Lecture 9-11 Sources of Magnetic Fields In 1820 Hans C. Oersted at University of Copenhagen observed that electric current creates magnetic field and lightning magnetizes iron. Since we could not find magnetic monopole, Gauss theorem for field gives: S ds q net 0 However a French contemporary of Oersted,Ampere, noted that an infinite straight current carrying wire creates a circulating magnetic field around the wire, where: 0 I 2 r r magnetic field circulates around wire. 0I 2 r where the magnetic permeability constant 0 is 7 N A also T m/ A

12 Lecture 9-12 In general: iot Savart Law A d field created by a dl element of a wire which conducts i current, at a distant point P from dl. d 0 idl r 3 4 r I Moving point charge (its of) current How do we calculate due to I? by using calculus, or Ampère s Law (sometimes)

13 Lecture 9-13 We have already discussed that a Circular Loop Current is a Magnetic Dipole front I 2r 0 ( at center) multiple loops

14 Lecture 9-14 Physics 219 Question 2 Sept A loop of wire lies in the x-y plane. This loop has a current I that circulates clockwise as viewed from above (from +z toward -z). What is the direction of the magnetic field at point A? z a) along +x b) along -y c) along +z d) along -z e) none of the above I A x y

15 Lecture 9-15 Physics 219 Question 3 Sept A loop of wire lies in the x-y plane. This loop has a current I that circulates clockwise as viewed from above (from +z toward -z). What is the direction of the magnetic field at point A? z a) along +x b) along -y c) along +z d) along -z e) none of the above A I x y

16 Lecture 9-16 Solenoid s field synopsis R L Long solenoid (R<<L): inside solenoid // to axis outside solenoid nearly zero (not very close to the ends or wires) Solenoid s field ar magnet s field

17 Lecture 9-17 Ampere s Law in Magnetostatics The sum of the product of (magnetic field projected along a path) and l (the path length) along a closed loop, Amperian loop, is proportional to the net current I net encircled by the loop, l loop l dl I 0 net l ( i i ) N A Choose a direction of summation. A current is positive if it flows along the RHR normal direction of the Amperian loop, as defined by the direction of summation.

18 Lecture 9-18 Calculation of the Solenoid Magnetic Field With the help of Ampere s Law: loop l I 0 net Long straight wire: Long solenoid: inside solenoid outside solenoid // to axis nearly zero r (not very close to the ends or wires) n windings per unit length 2 r I 0 0I 2r h ( nhi ) ni 0 0

19 Lecture 9-19 Two Parallel Currents 1 2 F I L I L L F 2 1 0I1 2 R II 2 R L F I I F L 2 R L Definition of charge 1 C (1 A for 1 Sec) The ampere is defined to be the constant parallel currents 1 m apart that will produce the force between them of 2 x 10-7 N per meter.