Velocity and acceleration vectors of a charged particle moving in a magnetic field at some instant are `vec(v)=3hat(i)+4hat(j)` and `vec(a)=2hat(i)+xhat(j)`. Selcet the wrong alternative.

Options

A. `x= -1.5`
B. x=3
C. Magnetic field is along z-direction
D. Kinetic energy of the particle is constant

Answer

A, C

Explanation

As `vec(F) bot vec(v)`, so `vec(F).vec(v)= 0 rArr m vec(a). vec(v)=0`
`rArr m(2 hat(i) + x hat(j)) .(3hat(i) + 4hat(j))= 0`
`rArr 6+ 4x = 0 rArr x= -1.5`
Let the magnetic field is `vec(B)= a hat(i) + b hat(j) + c hat(k)`, then from `vec(F) = q (vec(v) xx vec(B))`
`rArr m (2 hat(i) + x hat(j)) =q(3 hat(i) + 4hat(j)) xx (a hat(i) + b hat(j) + c hat(k))`
`rArr 2m hat(i) -1.5 m hat(j)= q(3b hat(k)- 3c hat(j)- 4a hat(k) + 4c hat(i))`
`:. 4c= 2m rArr c ne 0`
and `3b-4a= 0 rArr b= (4)/(3)a`
So, a and b both may or may not be zero.