A particle of charge q moves with velocity `vec(v)` along positive y - direction in a magnetic field `vec(B)`. Compute the Lorentz force experienced by the particle (a) when magnetic field is along positive y-direction (b) when magnetic field points in positive z -direction (c ) when magnetic field is in zy - plane and making an angle `theta` with velocity of the particle. Mark the direction of magnetic force in each case.

Explanation

Velocity of the particle is `vec(v) = v hat(j)`
(a) Magnetic field is along positive y-direction, this implies, `vec(B) = v hat(j)`
From Lorentz force, `vec(F_(m)) = q(v hat(j) xx B hat(j) ) ` = 0
So, no force acts on the particle when it moves along the direction of magnetic field.
(b) Magnetic field points in positive z - direction, this implies,
`vec(B) = B hat(K)`
From Lorentz force,
` vec(F_(m)) = q (v hat(j) xx B hat(k)) = "qvB"hat(i)`
Therefore, the magnitude,of the Lorentz force is qvB and direction is along positive x - direction .
(c ) Magnetic field is in zy - plane and making an angle `theta` with the velocity of the particle , which implies
`vec(B) " B cos" theta aht(j) + ` B sin `theta hat(k)`
From Lorentz force,
`vec(F_(m)) = q(v hat(j)) xx (B cos theta hat(j) + B sin theta hat(k))`
= qvBsin `theta hat(i)`