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 \((B=0.50 \mathrm{~T})\) and a muon enters this field with a velocity of \(3.0 \cdot 10^{6} \mathrm{~m} / \mathrm{s}\) at a right angle to the field. What will be the radius of the resulting orbit of the muon?

Short answer

In this exercise, a muon particle with mass \(1.88 \times 10^{-28}\text{ kg}\) and velocity \(3.0 \times 10^6\text{ m/s}\) enters a magnetic field of strength \(0.50 \text{ T}\) at a right angle. Calculate the radius of the resulting orbit. The muon's orbit radius is approximately \(9.47\times 10^{-3}\text{ m}\) or \(9.47\text{ mm}\).

Step-by-step solution

Calculate the magnetic force on the muon

Since the muon enters the magnetic field at a right angle, the force acting on it due to the magnetic field can be calculated using the formula: \(F = qvB\sin(\theta)\) As the muon has an identical charge as that of an electron, \(q=-1.60 \times 10^{-19} \text{C}\) and given \(\theta=90^{\circ}\) and \(\sin(90^{\circ})=1\), so: \(F = (-1.60 \times 10^{-19})(3.0 \times 10^6)(0.50)\) \(F = 2.40 \times 10^{-13} \text{N}\)

Equate the magnetic force to the centripetal force

The magnetic force acting on the muon is equal to the centripetal force required to keep the muon in an orbit: \(F = \frac{mv^2}{r}\) Where \(m=1.88 \times 10^{-28}\text{ kg}\), \(v=3.0 \times 10^6\text{ m/s}\) and \(r\) denotes the radius. We can substitute the values and equate the two forces: \(2.40 \times 10^{-13} = \frac{(1.88 \times 10^{-28})(3.0 \times 10^6)^2}{r}\)

Calculate the radius of the muon's orbit

Now, we will solve for the radius \(r\): \(r=\frac{(1.88 \times 10^{-28})(3.0 \times 10^6)^2}{2.40 \times 10^{-13}}\) \(r = 9.47 \times 10^{-3} \text{m}\) So the radius of the resulting orbit of the muon is \(9.47\times 10^{-3}\text{ m}\) or \(9.47\text{ mm}\).

Key concepts

Muon

Muons are fascinating subatomic particles that are similar to electrons in many ways. They have the same charge as electrons, which is

• negative and equal to \(-1.60 \times 10^{-19}\) Coulombs,
• but muons are significantly heavier than electrons, with a mass of \(1.88 \times 10^{-28}\) kilograms.

This makes muons much more massive, which affects how they interact with magnetic fields.

Muons are abundant in cosmic rays, particles from outer space that collide with the Earth's atmosphere. In experiments, such as the one described in our exercise, muons can be manipulated using magnetic fields to learn more about their properties and behavior.

Centripetal Force

Centripetal force is essential for understanding the circular motion of objects, like the orbit of a muon in a magnetic field.

Centripetal force is the inward force required to keep an object moving in a circular path, always pointing towards the center of the circular path.

• It can be calculated using the formula \(F = \frac{mv^2}{r}\),
• where \(m\) is the mass of the object,\(v\) is its velocity, and\(r\) is the radius of the circular path.

In our scenario, the magnetic force acting on the muon provides the necessary centripetal force to maintain its circular orbit.

This means the forces acting on the muon are in perfect balance, allowing it to travel smoothly along its curved trajectory.

Magnetic Field

Magnetic fields are invisible forces that can exert influence on charged particles, making them follow curved paths.

In this exercise, a strong magnetic field (0.50 T) is used to bend the path of the muon.

• The interaction of the magnetic field with the muon's charge results in a magnetic force,
• calculated using the formula\(F = qvB\sin(\theta)\).

Since the muon enters the magnetic field at a right angle, \(\theta = 90^{\circ}\) and \(\sin(90^{\circ}) = 1\), simplifying the equation.

The concept of the magnetic force is crucial because it is exactly this force that keeps the muon in its circular orbit by providing the centripetal force required to maintain circular motion.