An electron and a position are released from (0, 0, 0) and `(0, 0, 1*5R)` respectively, in a uniform magnetic field `vecB=B_0hati`, each with an equal momentum of magnitude `p=eBR`. Under what conditions on the direction of momentum will the orbits be non-intersecting circles?

Options

A. ` cos theta lt 3/8`
B. `2 cos theta lt 1/8`
C. `cos theta gt 1/8`
D. `cos theta gt 3/8`

Answer

A

Explanation

Ans B is along the x-axis, for a circular orbit the momenta of the two particles are in the y- z plane. Let `P_1` and `P_2` be the momentum of the electron and position, respectively. Both of them define a circle of radius R. They shall define circles of opposite sense. Let` P_1` make and angle ` theta ` with the y -axis, `P_2` must make the same angle. The centre of the respective circle must be perpendicular to the momenta and at `C_p`
`(0 - R sin theta , 3/2 R - R cos theta) `
The circle of the two shall not overlap if the distance between the two centres are greater than 2R. Let d be the distance between `C_p` and `C_e`

Then `d^2 = (2R sin theta)^2 + (3/2 - 2R cos theta)^2`
` = 4R^2 sin^2 theta + 9/4 R^2 - 6R^2 cos theta + 4R^2 cos^2 theta `
` = 4R^2 + 9/2 R62 - 6R^2 cos theta `
Sinced d has to be greater than `2R d^2 gt 4R^2`
` rArr 4R^2 + 9/2 R^2 - 6R^2 cos theta gt 4R^2`
` rArr 9/2 gt 6 cos theta " or " cos theta lt 3/8`