Q25 of 29 Page 26

An electron and a positron are released from (0, 0, 0) and (0, 0, 1.5R ) respectively, in a uniform magnetic field , each with an equal momentum of magnitude p = e BR. Under what conditions on the direction of momentum will the orbits be nonintersecting circles?


Because of electric charge the initial velocities are along –z for positron and +z for electron. Then it experiences Lorentz force and start rotation. The rotation is limited to the y-z plane as no involvement of any x directing components.


The momentum P is always perpendicular to the radius in circular motion.



Let us assume the momentum of the electron is Pelectron and the positron is Ppositron.


So, the coordinates of the center of their paths are going to be,


Ce = (0, Rsinθ, Rcosθ)


Cp = (0, -Rsinθ, 1.5R-Rcosθ)


The condition for the paths being non intersecting circles is that the distance between the centers is greater than 2R.


Distance between the center,


D2 = (Rsinθ - (-Rsinθ)2 + (Rcosθ - (1.5R – cosθ))2


= 4R2sin2θ + (2Rcosθ – 1.5R)2


= 4R2sin2θ + 4R2cos2θ – 6R2cosθ +2.25R � � � �2


= 6.25R2 – 6R2cos2θ


From the condition; D2>4R2


6.25R2 – 6R2cos2θ > 4R2


2.25/6 > cosθ


cosθ < 0.375


More from this chapter

All 29 →
23

A 100 turn rectangular coil ABCD (in XY plane) is hung from one arm of a balance (Fig. 4.4). A mass 500g is added to the other arm to balance the weight of the coil. A current 4.9 A passes through the coil and a constant magnetic field of 0.2 T acting inward (in xz plane) is switched on such that only arm CD of length 1 cm lies in the field. How much additional mass ‘m’ must be added to regain the balance?


 24

A rectangular conducting loop consists of two wires on two opposite sides of length l joined together by rods of length d. The wires are each of the same material but with cross-sections differing by a factor of 2. The thicker wire has a resistance R and the rods are of low resistance, which in turn are connected to a constant voltage source V0. The loop is placed in uniform a magnetic field B at 45° to its plane. Find τ, the torque exerted by the magnetic field on the loop about an axis through the centres of rods.


 26

A uniform conducting wire of length 12a and resistance R is wound up as a current carrying coil in the shape of (i) an equilateral triangle of side a; (ii) a square of sides a and, (iii) a regular hexagon of sides a. The coil is connected to a voltage source V0. Find the magnetic moment of the coils in each case.

 27

Consider a circular current-carrying loop of radius R in the x-y plane with center at origin. Consider the line integral

taken along z-axis.


(a) Show that monotonically increases with L.


(b) Use an appropriate Amperian loop to show that .