Question

A dipole of fixed length 2R has mass m on each end, charge +Q2 on one end and Q2 on the other. It is in orbit around a fixed point charge +Qı. (The ends of the dipole are constrained to remain in the orbital plane.) Figure 1.57 shows the definitions of the coordinates r, 0, a. Figure 1.58 gives the radial distances of the dipole ends from +Q1. (a) Using the Lagrangian formulation, determine the equations of mo tion in the (1,0, a) coordinate system, making the approximation r » R when evaluating the potential. (b) The dipole is in a circular orbit about Qı with ræözöæ0 and a< 1. Find the period of small oscillations in the a coordinate. (Wisconsin) 2R RA 02 Q1 Fig. 1.57 Fig. 1.58

Answer 1

2R V Y r the (a) The angle axis the angular bo i polan its centre between the dipole and is (0+2) dipole about (6+2) of the velocity is of mass is of the dipole > 2 d. kinetic energy T = in the 1 (2m). (82 + 2 8 2) + 12. 2m R 2 m R² Corås mie + m (m2+ R²) 6 ² + m R² à ² + 2 mkoa placed electric field charge Q,, so potential dipole is ~ Since the dipole is of of the energy O, Q2 V ། 47 to LA Vita & cosa Vr²+R N r+R+ 28 R cosa + 22 + 2. 2 R cosa Nr 1 1. 2 ? cosa 2 r r = r I R cosa. 31- r. r+ R cosa r- Rcosa

하 SH 2 R cosa 2 – R² costa + 2 R cos of 72 and the potential energy is VE 2 2 R cosa 40 ton re -> is Lagrangian L = T - V m² + m (r² + R²) 0 ² + m R² 22 + 2 m Roa Q, Q2 2Rcosa 40€ 22 + 72 Lagrange's equation gives (8) - 2 = 0 d at 2 C (emr) - 2 mro + 28, Q2 2Rcosa = 0. 4Xto 73 mi 2 R cosa + 2 mno2+Q, & 40€ (;) r3 Other one gives y #) - = 0; = 0 2m (22+R²) + 2m R² i +2mrros (می) - 2L IL and = 0 & CO gives dt aa Q,Q, sind mR (ätö ) + = 0 – (on) 40 to 82

are the equations of motion Equations (i), (i), (2) dipole. of the is a constant * * * 0. As 15$ (6) Also, = 0 <<1 (ie. sində a) from (ui) MR ä + Q, Q, 30, Anto 82 in a is motion the that This shows angular with a frequency simple harmonic Q,Q2 47€ MR82 is The period of such oscillation 2T 2T Anto mRr2 Q, Q2

Here is the equation of motion and time period for the given system.