Animated Solution for Physics - Magnetic Effects of Current: ${\text{H}}^{+}$, ${\text{He}}^{+}$ and ${\text{O}}^{2+}$ all having the same kinetic energy pass through a region in which there is a uniform magnetic field perpendicular to their velocity. The masses of ${\text{H}}^{+}$, ${\text{He}}^{+}$ and ${\text{O}}^{2+}$ are $1\text{ amu}$, $4\text{ amu}$ and $16\text{ amu}$ respectively. Then

The problem of charged particles moving through a magnetic field is a classic in physics, beautifully combining kinematics with electromagnetism. When a charged particle enters a uniform magnetic field perpendicular to its velocity, it experiences a magnetic force that acts as a centripetal force, bending its path into a circle.

The radius $r$ of this circular path is determined by the balance between the magnetic force and the required centripetal force:

However, the problem gives us a crucial constraint: all particles have the same kinetic energy $K$. To make our lives easier, we need to express the momentum $p=mv$ in terms of $K$.

Recall the relationship between kinetic energy and momentum:

Substituting this into our radius equation, we get a much more useful form:

Since the kinetic energy $K$ and the magnetic field $B$ are identical for all three particles (${\text{H}}^{+}$, ${\text{He}}^{+}$, and ${\text{O}}^{2+}$), the radius of their paths depends entirely on their mass $m$ and charge $q$. We can write a proportionality relation:

Now, let's calculate this ratio for each particle.

1. The Proton (${\text{H}}^{+}$): It has a mass of $1\text{ amu}$ and a charge of $+1e$.

2. The Helium Ion (${\text{He}}^{+}$): It has a mass of $4\text{ amu}$ and a charge of $+1e$.

3. The Oxygen Ion (${\text{O}}^{2+}$): It has a mass of $16\text{ amu}$ and a charge of $+2e$.

Comparing the ratios, we find:

The proton (${\text{H}}^{+}$) has the smallest radius. In the context of circular motion, a smaller radius means the path curves more sharply. Therefore, the proton experiences the greatest change in direction, meaning it is deflected the most.

On the other hand, the helium and oxygen ions have the exact same radius, which is twice as large as the proton's. Because their radii are identical, they will follow the exact same path and be deflected equally.