Understanding Charged Particle Motion in a Magnetic Field

When a charged particle enters a uniform magnetic field perpendicular to its velocity, the magnetic force acts as a centripetal force, causing the particle to move in a circular trajectory. The radius of this circular path depends on the particle's mass, charge, velocity, and the strength of the magnetic field.

Formula for Radius of Trajectory

The magnetic force on a charged particle moving perpendicular to a magnetic field is given by:

\(F_B = qvB\)

This force provides the centripetal force required for circular motion:

\(F_c = \frac{mv^2}{r}\)

Equating the two forces, we get:

\(qvB = \frac{mv^2}{r}\)

Solving for the radius \(r\), we find:

\(r = \frac{mv}{qB}\)

Relating Radius to Kinetic Energy

The kinetic energy (KE) of a particle is given by:

\(KE = \frac{1}{2}mv^2\)

From this, we can express velocity \(v\) in terms of KE and mass \(m\):

\(v^2 = \frac{2KE}{m}\)

\(v = \sqrt{\frac{2KE}{m}}\)

Substitute this expression for \(v\) into the radius formula \(r = \frac{mv}{qB}\):

\(r = \frac{m}{qB} \sqrt{\frac{2KE}{m}}\)

We can rewrite the mass \(m\) as \(\sqrt{m^2}\) inside the square root:

\(r = \frac{1}{qB} \sqrt{\frac{m^2 \cdot 2KE}{m}}\)

\(r = \frac{1}{qB} \sqrt{2mKE}\)

This formula shows that the radius of the trajectory is proportional to \(\frac{\sqrt{m}}{q}\) when the kinetic energy (KE) and magnetic field (B) are constant.

Properties of Proton and Alpha Particle

We need to compare a proton and an alpha particle. Let's list their relevant properties:

Calculating the Ratio of Radii

The problem states that the proton and alpha particle have the same kinetic energy, let's call it KE. They also enter the same uniform magnetic field \(B\). Using the formula \(r = \frac{\sqrt{2mKE}}{qB}\):

Radius of proton's trajectory (\(r_p\)):

\(r_p = \frac{\sqrt{2m_p KE}}{q_p B} = \frac{\sqrt{2m_p KE}}{eB}\)

Radius of alpha particle's trajectory (\(r_\alpha\)):

\(r_\alpha = \frac{\sqrt{2m_\alpha KE}}{q_\alpha B}\)

Substitute \(m_\alpha = 4m_p\) and \(q_\alpha = 2e\):

\(r_\alpha = \frac{\sqrt{2(4m_p) KE}}{2eB} = \frac{\sqrt{8m_p KE}}{2eB}\)

\(r_\alpha = \frac{\sqrt{4 \cdot 2m_p KE}}{2eB} = \frac{2\sqrt{2m_p KE}}{2eB} = \frac{\sqrt{2m_p KE}}{eB}\)

Now, let's find the ratio of the radii \(r_p : r_\alpha\):

\(\frac{r_p}{r_\alpha} = \frac{\frac{\sqrt{2m_p KE}}{eB}}{\frac{\sqrt{2m_p KE}}{eB}}\)

\(\frac{r_p}{r_\alpha} = 1\)

So, the ratio of the radii of their trajectories is 1:1.

Summary of the Radius Ratio Calculation

We used the formula for the radius of a charged particle's trajectory in a perpendicular magnetic field, expressed in terms of kinetic energy:

\(r \propto \frac{\sqrt{m}}{q}\) (when KE and B are constant)

For a proton:

\(m_p = m, q_p = e\)

\(r_p \propto \frac{\sqrt{m}}{e}\)

For an alpha particle:

\(m_\alpha = 4m, q_\alpha = 2e\)

\(r_\alpha \propto \frac{\sqrt{4m}}{2e} = \frac{2\sqrt{m}}{2e} = \frac{\sqrt{m}}{e}\)

Comparing the proportionality for \(r_p\) and \(r_\alpha\), we see they are the same. Therefore, the ratio \(r_p : r_\alpha\) is 1:1.

Revision Table: Key Formulas

Additional Information on Charged Particles in Magnetic Fields

Understanding how charged particles behave in magnetic fields is fundamental in physics. Here are some related points:

• If the velocity of the charged particle is not perpendicular to the magnetic field, the trajectory will be helical instead of purely circular. The component of velocity parallel to the field remains constant, while the perpendicular component causes circular motion.
• The frequency of the circular motion (known as the cyclotron frequency) is given by \(\omega = \frac{qB}{m}\) or \(f = \frac{qB}{2\pi m}\). Notice that this frequency is independent of the particle's velocity or kinetic energy.
• Devices like cyclotrons and mass spectrometers utilize the principles of charged particle motion in magnetic fields. Cyclotrons accelerate particles using an oscillating electric field while bending their path with a magnetic field. Mass spectrometers use magnetic (and sometimes electric) fields to separate particles based on their mass-to-charge ratio (\(\frac{m}{q}\)).
• The direction of the magnetic force is given by the right-hand rule (for positive charges) or left-hand rule (for negative charges), being perpendicular to both the velocity vector and the magnetic field vector.