Question

The Earth is showered with particles from space known as muons. They have a charge identical to that of an electron but are many times heavier \(\left(m=1.88 \cdot 10^{-28} \mathrm{~kg}\right)\). Suppose a strong magnetic field is established in a lab \((B=0.500 \mathrm{~T})\) and a muon enters this field with a velocity of \(3.00 \cdot 10^{6} \mathrm{~m} / \mathrm{s}\) at a right angle to the field. What will be the radius of the muon's resulting orbit?

Short answer

The charge of the muon is identical to that of an electron. Answer: The radius of the muon's orbit is approximately \(7.05 \times 10^{-3}\;\mathrm{m}\).

Step-by-step solution

Identify the given values

We are given the following values: Mass of the muon, \(m = 1.88 \times 10^{-28} \mathrm{~kg}\) Velocity of the muon, \(v = 3.00 \times 10^{6} \mathrm{~m/s}\) Magnetic field strength, \(B = 0.500 \mathrm{~T}\) Charge of the muon, \(q = -1.60 \times 10^{-19} \mathrm{~C}\) (since it's identical to the charge of an electron)

Use the orbit radius formula

To calculate the radius of the muon's orbit, we can use the formula: \(r = \frac{mv}{qB}\)

Substitute the given values and calculate the radius

Now, we can substitute the given values into the formula: \(r = \frac{(1.88 \times 10^{-28}\text{ kg})(3.00 \times 10^6\text{ m/s})}{(-1.60 \times 10^{-19}\text{ C})(0.500\text{ T})}\)

Solve the equation

After substituting the numbers, we can simplify the expression to find the radius: \(r = \frac{5.64 \times 10^{-22}}{-8.00 \times 10^{-20}}\) \(r \approx 7.05 \times 10^{-3}\;\mathrm{m}\) So, the radius of the muon's resulting orbit is approximately \(7.05 \times 10^{-3}\;\mathrm{m}\).

Key concepts

Charged Particle Motion

The motion of charged particles, such as muons, is deeply rooted in the laws of electromagnetism. Whenever a charged particle moves through a magnetic field, it experiences a force that acts perpendicular to both its velocity and the magnetic field. This force doesn't do any work on the particle - it doesn't increase or decrease the particle's energy - but it can change the direction of the particle's velocity. Consequently, if a particle moves into a magnetic field perpendicularly (as with our muon), it follows a circular path.

This phenomenon holds true for all charged particles in magnetic fields, from electrons in cathode ray tubes to protons in the Large Hadron Collider. Understanding this concept allows us to predict and calculate the path of particles in various scientific and technological applications, exemplified by the muon's orbit in our exercise.

Magnetic Force on Moving Charge

The magnetic force on a moving charge is given by the equation \(\mathbf{F} = q(\mathbf{v} \times \mathbf{B})\), where \(q\) is the charge of the particle, \(\mathbf{v}\) is its velocity vector, and \(\mathbf{B}\) is the magnetic field vector. The force is maximal when the charge moves perpendicular to the magnetic field, leading to a circular path called 'cyclotron motion'.

Significance of the Right Angle

In our exercise, the muon enters the magnetic field at a right angle, meaning the angle between \(\mathbf{v}\) and \(\mathbf{B}\) is 90 degrees. This results in the maximum possible magnetic force and, consequently, a uniform circular motion. The direction of the force is determined by the right-hand rule, a useful mnemonic to understand the vector nature of electromagnetism.

Classical Mechanics in Physics

Classical mechanics is the branch of physics that deals with the motion of objects. It is governed by Newton's laws of motion and is applicable to a wide range of macroscopic scenarios. The principle used to solve our muon problem is part of classical mechanics, even though it involves electromagnetic force.

In this context, we consider two main concepts: inertia, which keeps the muon moving in a straight line at a constant speed, and the centripetal force, required for circular motion, provided by the magnetic force. The balance between these forces determines the muon's path in terms of its radius, highlighting the interconnectedness between classical mechanics and electromagnetism.

Cyclotron Motion

Cyclotron motion refers to the spiral path that a charged particle follows within a magnetic field when not entering perpendicularly. In a uniform magnetic field and absent other forces, a charged particle moving perpendicularly will follow a circular path known as cyclotron motion. Its radius is determined by the particle's speed, mass, charge, and the magnetic field's strength, as illustrated in our exercise.

The concept of cyclotron motion lays the foundation for the cyclotron machine, where charged particles are accelerated and their paths are bent using magnetic fields, demonstrating practical applications of circular motion of charged particles in controlled settings. In the case of our muon, the precise calculation of its orbit showcases these underlying principles in a straightforward example.