Problem 2 a ph o 2
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1) The motion of an electric dipole in a magnetic field is coupled between translation and rotation. The conservation laws for momentum and angular momentum are modified. 2) For a dipole of two charges connected by a rigid rod, the conserved quantities are the total momentum P and energy E, which are functions of the center of mass velocity and angular velocity. 3) The quantity J, defined as the sum of the cross product of the center of mass position and total momentum with the angular momentum, is also conserved.
Problem 2 a ph o 2
- 1. APHO II 2001 Theoretical Question 2 2013/09/21 04:59 A9/P9 p. 1 / 2 Theoretical Question 2 Motion of an Electric Dipole in a Magnetic Field In the presence of a constant and uniform magnetic field B , the translational motion of a system of electric charges is coupled to its rotational motion. As a result, the conservation laws for the momentum and the component of the angular momentum along the direction of B are modified from the usual form. This is illustrated in this problem by considering the motion of an electric dipole made of two particles of equal mass m and carrying charges q and q− respectively (q > 0). The two particles are connected by a rigid insulating rod of length , the mass of which can be neglected. Let 1r be the position vector of the particle with charge q, 2r that of the other particle and = 1r - 2r . Denote by ω the angular velocity of the rotation around the center of mass of the dipole. Denote by CMr and CMv the position and the velocity vectors of the center of mass, respectively. Relativistic effects and effects of electromagnetic radiation can be neglected. Note that the magnetic force acting on a particle of charge q and velocity v is vq × B , where the cross product of two vectors 1A × 2A is defined, in terms of the x, y, z, components of the vectors, by ( 1A × 2A )x ＝ ( 1A )y ( 2 A )z － ( 1A )z ( 2A )y ( 1A × 2A )y ＝ ( 1A )z ( 2A )x － ( 1A )x ( 2A )z ( 1A × 2A )z ＝ ( 1A )x ( 2A )y － ( 1A )y ( 2A )x. (1) Conservation Laws (a) Write down the equation of motion for the center of mass of the dipole by computing the total force acting on the dipole. Also, write down the equation of motion for the rotation about the center of mass by computing the total torque on the dipole about its center of mass. (b) From the equation of motion for the center of mass, obtain the modified form of the conservation law for the total momentum. Denote the corresponding modified conserved quantity by P . Write down an expression in terms of CMv and ω for the conserved energy E. 1
- 2. APHO II 2001 Theoretical Question 2 2013/09/21 04:59 A9/P9 p. 2 / 2 (c) The angular momentum consists of two parts: One part is due to the motion of the center of mass and the other is due to rotation around the center of mass. From the modified form of the conservation law for the total momentum and the equation of motion of the rotation around the center of mass, prove that the quantity J, defined below, is conserved. J = ( CMr × P ＋Iω )． Bˆ where Bˆ is the unit magnetic field vector. Note that 1A × 2A ＝ 2A − × 1A 1A ‧( 2A × 3A ) ＝ ( 1A × 2A )‧ 3A 321231321 )()()( AAAAAAAAA ⋅−⋅=×× for any three vectors 1A , 2A and 3A . Repeated application of the above first two formulas may be useful in deriving the conservation law in question. In the following, let B be in the z-direction. (2) Motion in a Plane Perpendicular to B Suppose initially the center of mass of the dipole is at rest at the origin, points in the x- direction and the initial angular velocity of the dipole is zˆ0ω ( zˆ is the unit vector in the z- direction). (a) If the magnitude of 0ω is smaller than a critical value cω , the dipole will not make a full turn with respect to its center of mass. Find cω . (b) For cases where 0ω > 0, what will be the maximum distance md in the x-direction that the center of mass can reach? (c) What is the total tension on the rod? Express it as a function of the angular velocity ω . 2