Animated Solution for Physics - Magnetic Effects of Current: A proton and an alpha particle, after being accelerated through same potential difference, enter uniform magnetic field, the direction of which is perpendicular to their velocities. Find the ratio of radii of the circular paths of the two particles.

Entering the Magnetic Arena

Imagine a proton and an alpha particle standing at the starting line of a particle accelerator. They are both subjected to the exact same accelerating potential difference, $V$. As they speed up, they gain kinetic energy and shoot out into a region filled with a uniform magnetic field, $\vec{B}$, which is perfectly perpendicular to their direction of motion.

When a charged particle enters a magnetic field perpendicularly, it experiences a magnetic force that acts as a centripetal force, causing it to move in a perfect circle. Our goal is to find the ratio of the radii of the circular paths they trace out.

Radius of the Path

The fundamental equation governing the radius $r$ of a charged particle moving in a magnetic field is:

Here, $m$ is the mass, $v$ is the velocity, $q$ is the charge, and $B$ is the magnetic field strength. However, we have a slight problem: we don't know their velocities directly. We only know they were accelerated through the same potential difference $V$.

Accelerating Potential

To bridge this gap, we need to relate the velocity $v$ to the accelerating potential $V$. The work done by the electric field is converted entirely into the particle's kinetic energy:

By rearranging this equation, we can solve for the velocity:

Now, let's substitute this expression for velocity back into our master radius equation:

By bringing the $m$ and $q$ from the denominator inside the square root, we get a beautiful, unified expression:

Comparing the Radii

Look closely at this new equation. The problem states that both the potential difference $V$ and the magnetic field $B$ are identical for both particles. Therefore, all the terms in the equation are constant except for the mass $m$ and the charge $q$. This gives us a powerful proportionality:

This means the ratio of their radii will simply be:

Now, we just need to plug in the properties of our particles. A proton has a mass $m$ and a charge $e$. An alpha particle is a helium nucleus, meaning it consists of two protons and two neutrons. Thus, its mass is roughly $4m$ and its charge is $2e$.

Substituting these values into our ratio:

The $m$ and $e$ terms cancel out beautifully, leaving us with:

And there we have it! The radius of the proton's path is $\frac{1}{\sqrt{2}}$ times the radius of the alpha particle's path.