Question: How long would it take a superparamagnetic particle to move from one sidewall of the microchannel to another? The motion of a superparamagnetic particle with magnetic moment mbead = PV (Mo + Ahead PHO can be described using the following equation (egn. 0): pVa = -6mnRU + PV(Mo . V)B + VXhead (B . V)B (0) Ho By solving this eqn. 0 we can find that the position of the particle inside the channel, x [m], can be described using egn. 1: dx dt =ax"+ Bx-3 (1) Eqn. 1 can be solved analytically to yield the dependence of the time (to, [s]) it would take a particle to move from one sidewall of the microchannel ( a = 104x10* [m]) to the other (b=73x10*[m]) on the current ( / [A]) passing through the wire (eqn. 2): B ( b' - a ? ) + = ( b- a) - P In ab+B (2) 3a aa + B In eqns 1 and 2, a = - PM, Ruhl B = _ Xhead Ra ul, 12 -, p=1089 [kg m-3], R=3x10*[m], 18x-n B cap (b' - a 2 0 2 ( 6 7 - a ? ) + 6- ( b - a ) - B - in/ (2) 30 aa + BB B2 (b-a) ab+ B In 3a 2c 2 (2) aa+ B In eqns 1 and 2, a = -. PM, R Al B= _Xbend Raug 12 , p=1089 [kg m-3], R=3x10*[m], 18x n (b' - a' )- B (b' - a? )+. B (b-a)- In ab + B 3a (2) aa + B In eqns 1 and 2, a = - PM, RAI . B =_Xhead Rug12 , p= 1089 [kg m-3], R=3x10*[m], 1=10" [kg mid s"]], M. = 0.05 [Am? kg"]], Xhead = 0.36, /, = 4x x10"' [kg m s ? A-2]. 1. [20 points] Derive egn. 1 from egn. 0, using the expressions for the components of initial magnetization (Mox, Moy, Moz), components of magnetic field (Bx, By, B2) and various partial derivatives of the components of magnetic field (such as, -") calculated in Appendix of Shevkoplyas et al. Lab Chip, 7, 1294-1302, 2007. (You must produce a step-by-step derivation of the equation to receive full credit.) 2. [15 points] Derive expression for trap assuming that the initial magnetization is zero