What is the radius of the track of a \(2 \mathrm{MeV}\) proton in a field of \(1 \mathrm{~T}\) ?

Imagine a proton whirling around a track, much like a car zipping along a circular race track. This motion requires a force to keep the proton in its curved path, and that force is known as centripetal force. Centripetal force acts perpendicular to the direction of velocity of the moving object and towards the center of its circular path.

In the context of a charged particle, like a proton, moving in a magnetic field, this force is provided by the magnetic field itself. According to the principles of physics, when a proton moves perpendicular to a magnetic field, a magnetic force acts on it, which serves the same purpose as centripetal force in keeping it in a circular motion.

To delve into the math, the centripetal force (\( F_c \/Fc)) can be calculated by the formula: \[ F_c = \frac{mv^2}{r} \] where:

• \( m \/) is the mass of the proton,
• \( v \/) is its velocity, and
• \( r \/) is the radius of the circular path it takes.

Understanding this force is crucial as it is the key to determining various properties of the particle's motion, such as its radius of curvature and velocity, both of which are integral to numerous applications in physics and engineering.

When you have a proton zooming in a magnetic field, it possesses what's called kinetic energy—the energy it has due to its motion. To understand its circular motion, we need to connect this kinetic energy to the velocity of the proton.

We can use the equation \[ E = \frac{1}{2}mv^2 \] which ties together kinetic energy (\( E \/)), mass (\( m \/)), and velocity (\( v \/)). Here's what each term represents:

• \( E \/) is the kinetic energy of the proton,
• \( m \/) is the mass, and
• \( v \/) is the velocity we want to find.

By rearranging this formula, we can solve for the velocity: \[ v = \sqrt{\frac{2E}{m}} \]

This step is a crucial middleman in the journey to find the proton's path radius. The bigger the kinetic energy, the faster our proton speeds and the larger the circular track it requires to contain its motion.

Finally, let's link everything together to reveal the secret behind the radius of a proton's track in a magnetic field. The circular track's radius is heavily influenced by both the magnetic force and centripetal force, which, as we've discovered, are one and the same in this context.

By equating the centripetal force with the magnetic force exerted by the magnetic field (\( qvB \/)), we get a formula where we can solve for the radius (\( r \/)): \[ r = \frac{mv^2}{qvB} \] The variables include:

• \( m \/) as the proton's mass,
• \( v \/) as the velocity we calculated from kinetic energy,
• \( q \/) as the charge of the proton, and
• \( B \/) as the magnetic field strength.

From here, it is a matter of plugging in the values to solve for the radius. Understanding this relationship is vital, as it can help physicists in experiments and engineers in designing devices like cyclotrons and other particle accelerators.