Animated Solution for Physics - Magnetic Effects of Current: An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of radii $r_e,r_p,r_{\alpha }$ respectively, in a uniform magnetic field $B$. The relation between $r_e,r_p,r_{\alpha }$ is

Imagine an electron, a proton, and an alpha particle, all injected into a uniform magnetic field with the exact same kinetic energy. They will all start moving in circular orbits. But how do their radii compare? This is a classic physics problem that tests your ability to manipulate formulas and understand proportionalities.

First, let's recall the fundamental physics at play. When a charged particle moves perpendicularly through a uniform magnetic field, it experiences a Lorentz force that acts as a centripetal force, causing it to move in a circle.

We can write this balance of forces as:

Rearranging this to solve for the radius $r$, we get:

This equation tells us the radius if we know the velocity. However, the problem states that the particles have the same kinetic energy, not the same velocity.

We need to express the velocity $v$ in terms of the kinetic energy $K$. We know the standard formula for kinetic energy:

Solving for $v$, we find:

Now, let's substitute this expression for $v$ back into our radius formula:

Simplifying this by bringing the mass $m$ inside the square root, we arrive at our master equation for this specific scenario:

Look closely at this final equation. The kinetic energy $K$ and the magnetic field $B$ are identical for all three particles. The number 2 is just a constant. Therefore, the radius $r$ depends only on the mass $m$ and the charge $q$ of the particle. We can write this as a proportionality:

Now, let's evaluate this ratio for our three contenders:

1. The Electron ($e^{-}$): It has a tiny mass $m_e$ and a charge magnitude of $e$.

2. The Proton ($p^{+}$): It has a mass $m_p$ and a charge $e$.

3. The Alpha Particle (${\alpha }^{2+}$): An alpha particle is a helium nucleus, consisting of 2 protons and 2 neutrons. Its mass is approximately $4m_p$ and its charge is $2e$.

Let's simplify the alpha particle's ratio. The square root of 4 is 2, which perfectly cancels out the 2 in the denominator:

This is exactly the same ratio as the proton! Therefore, despite being four times heavier and having twice the charge, the alpha particle will trace out the exact same circular path as the proton.

Finally, what about the electron? The mass of an electron is roughly 1836 times smaller than the mass of a proton ($m_e\ll m_p$). Because its mass is so incredibly small, the numerator in its ratio is tiny, resulting in the smallest radius of the three.

Combining these findings, we get our final relationship:

Always pay close attention to the constraints in these problems. If the question had stated they had the same momentum instead of kinetic energy, the relationship would have been entirely different!