Problem 53 - Fundamentals of Physics

Question

In Module \(28-4,\) we showed that a particle of charge \(q\) and mass \(m\) will move in a circle of radius \(r=m v /|q| B\) when its velocity \(\vec{v}\) is perpendicular to a uniform magnetic field \(\vec{B}\). We also found that the period \(T\) of the motion is independent of speed \(v\) These two results are approximately correct if \(v

Step-by-Step Solution

Answer

(a) The correct period is \( T = \frac{2\pi \gamma m}{|q| B} \). (b) No, \( T \) is not independent of \( v \). (c) and (d) Calculate using given values and respective formulas. (e) and (f) Follow the same calculations for each case.

Step 1: Understanding the Relativistic Radius Formula

The formula for the radius of a particle in a magnetic field at relativistic speeds is given by \( r = \frac{\gamma m v}{|q| B} \), where \( \gamma \) is the Lorentz factor, defined as \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \). This formula accounts for the relativistic effects that are not present at lower speeds.

Step 2: Finding the Expression for the Period

The period \( T \) of the circular motion is given by the formula \( T = \frac{2\pi r}{v} \). Substituting the expression for \( r \) from the relativistic radius formula, we get:\[T = \frac{2\pi \left(\frac{\gamma m v}{|q| B}\right)}{v} = \frac{2\pi \gamma m}{|q| B}\]Thus, the correct expression for the period in terms of \( \gamma \), \( m \), \( |q| \), and \( B \) is independent of \( v \).

Step 3: Checking Independence of Period from Velocity

From the derived formula \( T = \frac{2\pi \gamma m}{|q| B} \), we observe that \( \gamma \) depends on \( v \). Therefore, as \( \gamma \) is a function of \( v \), the period \( T \) is not independent of the velocity \( v \) at relativistic speeds.

Step 4: Calculate the Radius According to Chapter 28

Using the non-relativistic formula \( r = \frac{mv}{|q| B} \) from Chapter 28, with the given kinetic energy \( E = 10.0 \text{ MeV} = 1.6 \times 10^{-12} \text{ J} \), we first find the velocity. The energy of an electron in relativistic terms is \( E = \gamma mc^2 - mc^2 \), and solving for \( v \), we can derive:\[v = c\sqrt{1 - \left(\frac{mc^2}{E + mc^2}\right)^2}\]Substitute \( m = 9.11 \times 10^{-31} \text{ kg} \), \( q = 1.6 \times 10^{-19} \text{ C} \), and \( B = 2.20 \text{ T} \) to find \( v \) and then \( r \).

Step 5: Calculate the Correct Radius

With \( p = \gamma mv \), use \( r = \frac{p}{|q| B} \) to calculate the correct radius with the relativistic speed and the same parameters as in the above step. This will give us the corrected value of \( r \) accounting for relativistic effects.

Step 6: Calculate the Period According to Chapter 28

The non-relativistic period is given by \( T = \frac{2\pi r}{v} \), where \( r \) is the radius from Step 4. Substitute the non-relativistic \( r \) to find \( T \).

Step 7: Calculate the Correct Period

The correct period is \( T = \frac{2\pi \gamma m}{|q| B} \) which includes the Lorentz factor \( \gamma \) and uses the correct radius found in Step 5. This gives us an accurate period accounting for relativistic speeds.

Key Concepts

Magnetic Fields

In physics, magnetic fields represent the invisible forces that arise from magnets and electric currents. Simply put, a magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. The strength of the magnetic field is denoted by the symbol \( B \) and is measured in teslas (T), which is the unit of magnetic induction.
Magnetic fields are fundamental to many aspects of physics and engineering. They play a crucial role in electromagnetism, one of the fundamental interactions of nature. When a charged particle, like an electron, moves through a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field. This force causes the particle to move in a circular path. Therefore, magnetic fields are essential for understanding the behavior of charged particles in a variety of applications, from simple magnets to complex devices like cyclotrons.

• Magnetic fields apply a force perpendicular to charged particle movement.
• Strength of field \( B \) is measured in teslas.
• They are key in technologies involving charged particles.

Lorentz Factor

The Lorentz factor, denoted by \( \gamma \) (gamma), is a crucial component in the study of relativistic physics. It's defined as \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \), where \( v \) is the velocity of an object, and \( c \) is the speed of light. This factor accounts for the relativistic effects that become significant when particles travel at speeds close to the speed of light.
As a particle's velocity approaches the speed of light, the Lorentz factor increases significantly, reflecting changes such as mass increase and time dilation. These effects are essential for accurately describing the behavior of particles at high speeds. In the context of a charged particle moving in a magnetic field, incorporating \( \gamma \) ensures that the calculated radius and period consider the relativistic effect.

• \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \) – accounts for relativistic effects.
• Important for speeds close to light speed.
• Affects calculations of radius and period in circular motion.

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle or a circular path. When a charged particle like an electron is placed perpendicular to a magnetic field, it moves in a circular path due to the magnetic force acting as the centripetal force.
The radius of this circular motion is given by the formula \( r = \frac{mv}{|q|B} \) for non-relativistic speeds. This formula is modified to \( r = \frac{\gamma mv}{|q|B} \) when accounting for relativistic speeds by including the Lorentz factor. Thus, the radius depends on the mass \( m \), velocity \( v \), and the charge \( q \) of the particle, along with the magnetic field \( B \). The period \( T \) of the motion is linked to the radius and the velocity by \( T = \frac{2\pi r}{v} \).

• Movement is due to a perpendicular magnetic force.
• Radius depends on velocity, charge, and magnetic fields.
• Period is calculated from radius and velocity.

Relativistic Effects

In the realm of physics, relativistic effects become noticeable when objects move at speeds comparable to the speed of light. At these high velocities, traditional Newtonian mechanics no longer applies, and Einstein's theory of relativity takes precedence.
Relativistic effects include time dilation, where time appears to slow down for fast-moving objects, and length contraction, where objects appear shorter in the direction of their motion. These effects are direct consequences of the finite speed of light and how time and space are interrelated.
For charged particles in a magnetic field, these effects mean that both the particle's mass and momentum increase significantly as the speed approaches that of light. Accurately accounting for these changes is crucial for precise calculations of motion paths and time cycles, which is why using the Lorentz factor \( \gamma \) is so important in the formulas for radius and period.

• Relativity applies at speeds close to light speed.
• Time dilation and length contraction are prominent effects.
• Changes in mass and momentum affect paths and periods.