Animated Solution for Physics - Magnetic Effects of Current: A particle of mass $m$ and charge $q$, moving with velocity $v$ enters Region II normal to the boundary as shown in the figure. Region II has a uniform magnetic field $B$ perpendicular to the plane of the paper. The length of the Region II is $l$. Choose the correct choice (s).

A Magnetic Boundary

Imagine you are a charged particle, zooming through space with a velocity $v$. Suddenly, you cross a boundary and enter a new region—Region II. But this isn't just empty space; it's a region filled with a uniform magnetic field $B$, pointing straight into the page.

What happens next is one of the most beautiful phenomena in physics. Because your velocity is perpendicular to the magnetic field, you experience a magnetic force. But this force doesn't speed you up or slow you down. Instead, it acts exactly perpendicular to your motion, pulling you sideways. This is the hallmark of a centripetal force, and it forces you into a perfect circular path.

The radius of this circular dance is a delicate balance between your momentum and the magnetic grip. It is given by the famous equation:

Here, $m$ is your mass, $v$ is your speed, $q$ is your charge, and $B$ is the magnetic field strength. The faster you go, or the heavier you are, the wider your circle. The stronger the field, or the higher your charge, the tighter the circle.

The Right-Hand Rule

Before we dive into the math, let's build a physical intuition for why the particle turns in the first place. When a charged particle moves through a magnetic field, it experiences the Lorentz force. The direction of this force is given by the right-hand rule (for a positive charge): point your fingers in the direction of velocity $v$, curl them towards the magnetic field $B$ (which is into the page), and your thumb points in the direction of the force.

In this case, as the particle enters Region II, the force pushes it upwards. But as the particle's velocity direction changes, the force direction changes with it, always remaining exactly perpendicular to the velocity.

Because the force is always perpendicular to the motion, it does zero work on the particle. This is a profound realization: a static magnetic field can never change the kinetic energy or the speed of a charged particle. It can only change its direction. The particle will move at a constant speed $v$ throughout its entire journey in Region II.

Entering Region III

Now, Region II isn't infinite. It has a strict width of $l$. Beyond it lies Region III. The ultimate question is: will you make it across, or will the magnetic field whip you around and send you back to Region I?

To cross into Region III, your circular path must be wide enough to breach the boundary at $x=l$. In mathematical terms, the radius of your path $r$ must be strictly greater than the width $l$.

Substituting our radius formula, we get:

Rearranging this for velocity, we find the critical condition:

If you are traveling faster than this critical speed, the magnetic field isn't strong enough to turn you around in time. You will burst through the boundary and enter Region III. This perfectly validates Option (a) and invalidates Option (b).

Maximum Path Length

But what if you travel at exactly the critical speed? What if $v=\frac{qBl}{m}$?

At this precise velocity, your radius $r$ becomes exactly equal to $l$. You will carve out a perfect semicircle. At the very peak of your arc, you will just graze the boundary of Region III, but you won't cross it. Instead, you will complete the semicircle and exit back into Region I.

This semicircular path of radius $l$ is incredibly special. It is the absolute longest path you can possibly take while remaining entirely within Region II. If you go any faster, you exit into Region III, cutting your journey short. If you go any slower, you complete a tighter semicircle, which has a smaller circumference.

Therefore, the path length is maximized when $v=\frac{qBl}{m}$. This confirms that Option (c) is absolutely correct.

Independent of Velocity

Finally, let's consider the time you spend in Region II, assuming you aren't fast enough to escape into Region III. If you return to Region I, you must have completed exactly half of a circle—a semicircle.

How long does it take to complete a full circle in a magnetic field? The time period $T$ is given by:

Since you only complete a semicircle, the time you spend in Region II is half of this period:

Look closely at this equation. Do you see a $v$ anywhere? No! The time spent in the magnetic field is completely independent of your velocity.

How is this possible? If you enter with a higher velocity, you travel along a larger circle, meaning you have a longer distance to cover. However, because you are moving faster, you cover that larger distance in the exact same amount of time. The increased distance perfectly cancels out the increased speed.

As long as you return to Region I, you will always spend the exact same amount of time in Region II, regardless of how fast you were going. This beautiful symmetry confirms that Option (d) is correct.

In conclusion, the correct choices are (a), (c), and (d). This problem is a masterful exploration of the dynamics of charged particles, testing your intuition on boundaries, path lengths, and the elegant independence of time in magnetic fields.