Question

Acceleration EMF (a) A long straight rod with cross sectional area A and conductivity σ accelerates parallel to its length with acceleration a. Write down the Drude-like equation of motion for the average velocity v of an electron of mass m in the rod relative to the motion of the rod itself. Show that, in the absence of an external electric field, there is a steady solution that corresponds to a current 1 Aơma/e flowing in the wire. Assume that a flexible, no-loss wire and a perfect ammeter close the circuit (b) Argue that this result generalizes to an EMF for the case of electrons in an accelerated closed circuit. Ignore the Drude drag force. (c) Use the results of part (b) to show that a circular wire ring rotated with angular acceleration 2 carries a current eR where R is the total resistance of the wire and S is the area of the wire circle. Estimate the magnitude of this current for a copper ring of radius 1 cm and cross sectional area 1 mm* which oscillates harmonically around the ring axis at 500 Hz, with a maximum angular ampitude of 1 of arc.

Answer 1

(a) From the Drude equation, total drag force acting on the electron depends on acceleration and relative velocity (v_r). F + F__re + mv/T = 0 ma + mv_r + mv/tau = 0 In the absence of electric field, the steady state conditions is, v_r = 0, ma + m(0)_r + mv/tau = 0 v = -a tau Mathematical expression for Drude-conductivity-equation is, sigma = ne^2 tau/m tau = m sigma/ne^2 Mathematical expression for current density is, J = nev = ne(a tau) = nea[m sigma/ne^2] = m sigma a/e Current in the wire is, l = JA = A sigma ma/e (b) If the electrons accelerated in the closed circuit, F_e = mv_r Ee = ma E = ma/e Generalizes to an EMF is, epsilon = contourintegral dl E = contourintegral dl (ma/e) (c) In duced current in the circuit is, I = epsilon/R = contourintegral dl (ma/e)/R Relation between linear and angular acceleration is,