Question

A proton (mass m and charge +e) and an alpha particle (mass 4m and charge +2e) are projected with the same kinetic energy at right angles to a uniform magnetic field.
Which one of the following statements will be true?

• A. the alpha-particle will be bent in a circular path with a smaller radius than that of the proton
• B. the radius of the path of the alpha-particle will be greater than that of the proton
• C. the alpha-particle and the proton will be bent in a circular path with the same radius
• D. the alpha-particle and the proton will go through the field in a straight line

Answer 1

The correct option is C the alpha-particle and the proton will be bent in a circular path with the same radius

Given than $K_p=K_a$

We know that $r=\frac{mv}{qB}=\frac{(2mK{)}^{1/2}}{qB}$

$\therefore r_p=\frac{(2m_p{K}_p{)}^{1/2}}{q_{p}B}$ and $r_{\alpha }=\frac{(2m_{\alpha }{K}_{\alpha }{)}^{1/2}}{q_{\alpha }B}$

Now $\frac{r_p}{r_{\alpha }}=\sqrt{\left(\frac{m_p}{m_{\alpha }}\right)}\times \frac{q_{\alpha }}{q_p}=\sqrt{\left(\frac{m_p}{4m_p}\right)}\times \frac{2e}{e}=1$