Physics I Class 23 Magnetic Force on Moving Charges Rev. 07-Apr-04 GB 23-1 Hendrick Antoon Lorentz Hendrick A. Lorentz was a Dutch physicist who refined certain aspects of electromagnetic theory. He, along with Irish mathematical physicist George F. FitzGerald, deduced fundamental properties of the electromagnetic behavior of moving bodies that formed the basis of Einstein’s Special Theory of Relativity. The force of a magnetic field on a moving charge is sometimes called the Lorentz Force. H.A. Lorentz 1853-1928 23-2 Important Facts About Velocity and Net Force Vectors (Review) v F Same direction: speeding up. v F Opposite directions: slowing down. v Right angles: changing direction, same speed. F 23-3 Vector Cross Product (Review) c a b ; | c | | a || b | sin( ) The direction comes from the right-hand rule. It is at a right angle to the plane formed by a and b . In other words, the cross product is at right angles to both a and b . (3D thinking required!) 23-4 Drawing 3D Vectors in 2D Y -Y +Y +X X -X Z -Z +Z +Z is out of page 23-5 Magnetic Force on a Moving Charge F q vB q: charge of the particle (C; + or –) v : velocity of the particle (m/s) B : magnetic field (T) Force is at a right angle to velocity. Force is at a right angle to magnetic field. Important: If q is negative, that reverses the direction of force. 23-6 An Example An Electron in a Magnetic Field Y X Z 23-7 Analysis of the Magnetic Force Y X Z F q vB We will evaluate this expression before F the electron starts turning. First, evaluate v B . In this case, they are 90° apart, so all we need is the direction. v is +X, B is –Z, so v B is +Y. Next, we need to account for q. This is an electron, so q is negative. Therefore, the magnitude of the force is (e v B) and the direction is –Y. 23-8 Uniform Circular Motion As the electron turns, so does the force vector. Speed stays constant because acceleration is always perpendicular to velocity. The electron travels in a circle at a constant speed. 23-9 The Radius of the Circle F v Although the directions of the vectors are changing, the magnitudes stay the same. r v2 F ma m r F qvB v2 qvB m r v2 mv rm qvB qB 23-10 The Period and Frequency F v The circumference of the circle is 2 r. Distance 2 r r v Time T mv 2 2r qB 2m T v v qB 1 qB f T 2m qB 2f m 23-11 Bubble Chamber The red and green lines in the figure to the left are tracks of charged particles in a bubble chamber. Each charged particle makes a trail of tiny bubbles as it moves in the chamber. There is a magnetic field of 1.0 T directed into the page. What are the signs of the charges of the particles? mv r Why do they spiral inward? qB What are they? What created them at the points where the tracks start? 23-12 The Aurora There is no acceleration in the direction of the magnetic field line. (Why?) The component of velocity in the direction of the field line remains constant. (Why?) The component of velocity at a right angle to the field line continually changes direction. (Why?) The result is that the charged particle (electron) travels in a spiral path along the magnetic field line, giving off light when it hits the atmosphere. 23-13 The Aurora As Seen from the Space Shuttle 23-14 The Effect of the Solar Wind on the Magnetic Field of Earth Energetic charged particles travel along magnetic field lines on the sun. Some escape into interplanetary space. These are called the solar wind. The solar wind interacts with the magnetic field lines of Earth and distorts them. The complex interaction of flowing charged particles with the electromagnetic field is called Magneto-Hydrodynamics or MHD. 23-15 Class #23 Take-Away Concepts 1. The magnetic (Lorentz) force on a moving, charged particle: F q vB 2. The magnetic force cannot change a particle’s speed, only the direction of its velocity. 3. Radius and angular frequency of a charged particle in uniform circular motion in a magnetic field: mv r qB qB m 23-16 Class #23 Problems of the Day ___1. A charged, non-magnetic particle is moving in a uniform magnetic field. Which of the following conditions (if any) would cause the particle to speed up? A) The velocity of the particle is at a right angle to the magnetic field. B) The velocity of the particle is in the same direction as the magnetic field. C) The velocity of the particle is in the opposite direction as the magnetic field. D) Any of the above (A-C) would cause the particle to speed up. E) None of the above; the magnetic force cannot cause the particle to speed up. 23-17 Class #23 Problems of the Day 2. An electron is traveling in a vacuum tube at 1.4 x 107 m/s in a horizontal direction toward the south. There is a constant magnetic field in the tube with a magnitude of 0.5 gauss. The direction of the magnetic field is toward the north and 30º down (toward the ground). What are the magnitude and direction of the magnetic (Lorentz) force on the electron? (1 T = 10,000 gauss.) Up 30° v S N -e B Down 23-18 Activity #23 Magnetic Field and Force Objective of the Activity: 1. Consider the implications of the magnetic force on speed and direction of a charged particle. 2. Determine the direction and magnitude of the magnetic field at your table in the classroom using a compass, a coil of wire, a power supply, and a current meter. 23-19

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