Question

The charged particles traverse identical helical paths in a completely opposite sense in a uniform magnetic field $B=B_0\hat{k}$.

• A. They have equal z-components of momenta
• B. They must have equal charges
• C. They necessarily represent a particle, antiparticle pair
• D. The charge to mass ratio satisfy
  ${\left(\frac{q_1}{m}\right)}_1+{\left(\frac{q_2}{m}\right)}_2=0$

Answer 1

The correct option is A They have equal z-components of momenta

As the two charged particles travers identical helical path the pitchof their helices will be same

Now pitch will be same

Now pitch $=$ Time for one circle completion $\times$ speed

$\Rightarrow P=T\times v\cos \theta =\frac{2\pi m}{qB}v\cos \theta$ (Time period under uniform magnetic field $=\frac{2\pi m}{qB})$

As $P_1=P_2$

$\Rightarrow \frac{2\pi m_1}{q_1{B}_0}v\cos \theta =\frac{2\pi m_2}{q_2{B}_0}v\cos \theta$

$\Rightarrow \frac{q_1}{m_1}=\frac{q_2}{m_2}$

So the have equal charge to mass eatio magnitude

But they are moving in opposite direction so one is $+Ve$ charge and another $-Ve$

$\Rightarrow \frac{q_1}{m_1}+\frac{q_2}{m_2}=0$