An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi

Q: Why do the particles complete exactly a semicircle?

Because the magnetic field is semi-infinite and they enter perpendicularly to the boundary, the magnetic force continuously turns them until their velocity vector points exactly opposite to the entry direction. At this point, they cross the boundary again and exit the field, having completed a $180^\circ$ turn, which is a semicircle.

Q: Why are the exit paths parallel?

Both particles enter the field travelling in the same direction (parallel to each other). Since they both undergo a $180^\circ$ turn inside the field, they will both exit travelling in the exact opposite direction to their entry. Therefore, their exit paths must also be parallel to each other.

Q: Does the magnetic field change their speed?

No. The magnetic force is always perpendicular to the velocity vector, meaning it does no work on the particles. It only changes their direction, not their kinetic energy or speed.

Q: Why does the proton have a larger radius?

The radius of the circular path is given by $r = \frac{mv}{qB}$. Since both particles have the same speed and charge magnitude, the radius is directly proportional to their mass. Because a proton is about 1836 times more massive than an electron, it has a much larger inertia and requires a larger radius to turn.

Q: Why do they exit at different times?

The time spent in the field is $t = \frac{\pi m}{qB}$. Even though the proton has a larger path to travel, its speed is the same as the electron's. Since time is distance divided by speed, the particle with the longer path (the proton) will take more time to exit.

Question

An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true?

Options:

(A) They will never come out of the magnetic field region

(B) They will come out travelling along parallel paths

(C) They will come out at the same time

(D) They will come out at different times

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine an electron and a proton, moving side-by-side like two racers on parallel tracks, entering a region where a uniform magnetic field exists. The field is semi-infinite, meaning it starts at a certain boundary and extends infinitely in one direction. The magnetic field is perpendicular to their velocity.

### The Magnetic Force and Circular Motion

When a charged particle enters a magnetic field perpendicularly, it experiences a magnetic force given by the Lorentz force equation:
$$ \vec{F}_m = q(\vec{v} 	imes \vec{B}) $$
This force is always perpendicular to both the velocity and the magnetic field, acting as a centripetal force that bends the particle's path into a circle.

By applying the right-hand rule, we can determine the direction of this force. For the positively charged proton, the force will push it in one direction (say, upwards). For the negatively charged electron, the force will be in the exact opposite direction (downwards).

Since the magnetic field is semi-infinite, the particles will complete exactly half a circle (a semicircle) before exiting the field region. Because they entered parallel to each other and both complete a $180^\circ$ turn, they will exit travelling in the exact opposite direction to their entry. Thus, they will emerge travelling along **parallel paths**.

### Comparing the Radii

The radius $r$ of the circular path is determined by balancing the magnetic force with the required centripetal force:
$$ qvB = \frac{mv^2}{r} \implies r = \frac{mv}{qB} $$
Since both particles have the same velocity $v$ and the same magnitude of charge $|q|$, the radius is directly proportional to their mass $m$:
$$ r \propto m $$
We know that the mass of a proton is significantly greater than the mass of an electron ($m_p \gg m_e$). Therefore, the proton will trace a much larger semicircle compared to the electron:
$$ r_p > r_e $$

### Time Spent in the Magnetic Field

Finally, let's determine how long each particle stays inside the magnetic field. The time spent is half of the time period $T$ of a full circular orbit:
$$ t = \frac{T}{2} = \frac{\pi m}{qB} $$
Just like the radius, the time spent in the magnetic field is directly proportional to the mass of the particle:
$$ t \propto m $$
Because the proton is much heavier than the electron, it will spend a proportionally longer time navigating its larger semicircular path:
$$ t_p > t_e $$
Therefore, they will **not** come out at the same time; they will emerge at different times.

### Conclusion

The electron and proton will both exit the magnetic field travelling along parallel paths, but the heavier proton will take longer to complete its journey.

Animated Solution for Physics - Magnetic Effects of Current: An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true?

Select Answer:

Visualized Solution

$Visualizing the Setup$

$Magnetic Force$

$Path of Particles$

$Exit Paths$

$Radius of Circular Path$

$Comparing Radii$

$Time Spent in Magnetic Field$

$Comparing Time Spent$

$Conclusion$

The Sigma Insight: Motion of a Charge in Magnetic Fields

Analyzing the Setup

The Magnetic Force and Circular Motion

Comparing the Radii

Time Spent in the Magnetic Field

Conclusion

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