Question

An electron and a proton are moving on straight parallel paths with the same velocity. They enter a semi-infinite region of a uniform magnetic field perpendicular to the velocity.

Which of the following statement(s) is/are true:

• A. They will never come out of the magnetic field region.
• B. They will come out traveling along parallel paths.
• C. They will come out at the same time.
• D. They will come out at different times.

Answer 1

The correct option is D

They will come out at different times.

Explanation for correct options:

In case of option (C),

1. The electron $\left(e\right)$ and proton $\left(p\right)$, both moving at the same velocity, enter the region of the uniform magnetic field.
2. The radius of the uniform circular path of a charged particle $\left(r\right)$ is given as,
   $r=\frac{mv}{qB}$, where $m$ is the mass of the particle, $v$ is the velocity of the particle, $q$ is the charge of the particle and $B$ is the magnetic field.
   In this case, we take charge of the particle $\left(q\right)$ as $e$. This is because we know that the charge of an electron is $-e$ and the charge of a proton is $+e.$
3. The figure gives us the path of the electron and proton in the magnetic field.
   From the figure below, we can see that both proton and electron enter the magnetic field perpendicularly and hence they will start circulating. But the direction of revolution will be opposite as they have opposite charges.
4. During the course of revolution, the charges will leave the magnetic field along parallel paths.

Therefore, we can say that both electron and proton will come out of the field along parallel paths.

In case of option (D),

1. We know that,

$$\begin{gathered} r=\frac{mv}{qB} \\ \Rightarrow r\propto m \\ \Rightarrow r_e<r_p\left[\because m_e<m_p\right] \end{gathered}$$

Where, $r_e$ is the radius of the electron, $r_p$ is the radius of the proton, $m_e$ is the mass of the electron and $m_p$ is the mass of the proton.

2. The radius of the path traversed by the electron will be less than that of the proton. Thus, the distance traveled by the proton will be more.

3. We know that the time period of revolution is given as,
$\text{time period=}\frac{\text{distance}}{\text{velocity}}$
$$\begin{gathered} \Rightarrow t=\frac{2\pi r}{v} \\ t=\frac{2\pi }{v}\left(\frac{mv}{qB}\right)\left[\because r=\frac{mv}{qB}\right] \\ t=\frac{2\pi m}{qB} \\ \Rightarrow t\propto m \\ \because m_e<m_p \end{gathered}$$
The time taken by the proton to leave the magnetic field will be more than that of the electron.

Thus, they will come out at different times.

Hence, options (B) and (D) are correct.