Animated Solution for Physics - Magnetic Effects of Current: Two particles $X$ and $Y$ having equal charges, after being accelerated through the same potential difference, enter a region of uniform magnetic field and describe circular paths of radii $R_1$ and $R_2$ respectively. The ratio of the mass of $X$ to that of $Y$ is

The journey of a charged particle through a magnetic field is one of the most elegant dances in physics. Imagine you are standing in a laboratory, observing two mysterious particles, $X$ and $Y$. You know they have the exact same electrical charge, but their masses are hidden from you. Your mission is to uncover the ratio of their masses using nothing but a voltage source and a magnet. Let's embark on this thrilling derivation!

Before the particles even see the magnetic field, they are subjected to an electric field. The problem tells us that both particles are accelerated through the same potential difference, $V$.

What does this mean physically? When a charge $q$ falls through a potential difference $V$, the electric field does work on it. By the work-energy theorem, this electrical work is entirely converted into the particle's kinetic energy ($K$).

Since both particles have the same charge $q$ and experience the same potential difference $V$, they must emerge with the exact same kinetic energy:

Now, these energized particles enter a region with a uniform magnetic field $B$. Because they are moving perpendicular to the field, the magnetic force acts as a centripetal force, bending their straight-line trajectories into perfect circles. Particle $X$ traces out a circle of radius $R_1$, and particle $Y$ traces out a circle of radius $R_2$.

To connect the radius of the path to the mass of the particle, we need our master equation for magnetic circular motion. The radius $R$ of a charged particle moving perpendicular to a magnetic field is given by the ratio of its momentum $p$ to the magnetic force factor $qB$:

But wait, we don't know their momenta directly; we only know their kinetic energies are equal. We need a bridge between momentum and kinetic energy. Recall the classic mechanics relation:

Let's substitute the kinetic energy $K=qV$ into this momentum equation:

Now, we bring this back to our radius formula. Substituting the momentum $p$ into the radius equation gives us a beautiful, comprehensive formula that links all our variables:

We can simplify this slightly by bringing the $q$ from the denominator inside the square root (where it becomes $q^2$):

This is where the magic happens. Look closely at our comprehensive radius equation. For both particle $X$ and particle $Y$, the values of $2$, $V$, $B$, and $q$ are absolutely identical. They are constants in this specific experiment.

When we strip away all the constants, we are left with a profound proportionality: the radius of the circular path is directly proportional to the square root of the particle's mass.

This means that a heavier particle will trace out a larger circle, assuming all other factors are equal. Now, let's write this proportionality as a ratio for our two particles:

We can combine the square roots on the right side:

The question asks for the ratio of the mass of $X$ to that of $Y$, which is exactly $\frac{m_X}{m_Y}$. To isolate this term, we simply square both sides of our equation. The square root vanishes, and we arrive at our final, elegant result:

And there we have it! By simply measuring the radii of their paths, we have successfully deduced the ratio of their hidden masses. This matches option (c). Physics is truly remarkable!