Problem 95 - Fundamentals of Physics

Question

Ionization measurements show that a particular lightweight nuclear particle carries a double charge \((=2 e)\) and is moving with a speed of \(0.710 c\). Its measured radius of curvature in a magnetic field of \(1.00 \mathrm{~T}\) is \(6.28 \mathrm{~m}\). Find the mass of the particle and identify it. (Hints: Lightweight nuclear particles are made up of neutrons (which have no charge) and protons (charge \(=+e\) ), in roughly cqual numbers. Take the mass of cach such particle to be 1.00 u.) (See Problem 53.)

Step-by-Step Solution

Answer

The particle is an alpha particle with mass approximately 0.40 u.

Step 1: Determine the Particle's Charge and Speed

We know that the particle carries a double charge, so its charge is \(q = 2e\) where \(e = 1.60 \times 10^{-19} \text{ C}\). The speed of the particle is given as \(0.710c\) where \(c = 3.00 \times 10^8 \text{ m/s}\). Thus, the speed of the particle is \(v = 0.710 \times 3.00 \times 10^8 = 2.13 \times 10^8 \text{ m/s}\).

Step 2: Use the Radius of Curvature Formula to Find Momentum

The formula relating the radius of curvature \(r\) in a magnetic field to momentum \(p\) is \(r = \frac{mv}{|q|B}\), where \(B = 1.00 \text{ T}\). Rearrange this to solve for the momentum: \(p = mv = |q|Br\). Substituting the values, \(p = (2 \times 1.60 \times 10^{-19} \text{ C})(1.00 \text{ T})(6.28 \text{ m}) = 2.01 \times 10^{-18} \text{ kg} \cdot \text{m/s}\).

Step 3: Calculate the Mass of the Particle

Now, apply the relativistic momentum equation \(p = \gamma mv \) where \(\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}\) is the Lorentz factor. First, calculate \(\gamma\):\(\gamma = \frac{1}{\sqrt{1 - (0.710)^2}} = \frac{1}{\sqrt{1 - 0.5041}} = \frac{1}{\sqrt{0.4959}} \approx 1.415\).Substitute back to find the mass:\[ m = \frac{p}{\gamma v} = \frac{2.01 \times 10^{-18} \text{ kg} \cdot \text{m/s}}{(1.415)(2.13 \times 10^8 \text{ m/s})} \approx 6.67 \times 10^{-28} \text{ kg}. \]

Step 4: Identify the Particle

Convert the mass from kilograms to atomic mass units (u), knowing that 1 u \(= 1.66 \times 10^{-27} \text{ kg}\). \[ m \approx \frac{6.67 \times 10^{-28} \text{ kg}}{1.66 \times 10^{-27} \text{ kg/u}} \approx 0.40 \text{ u}. \]The calculated mass is approximately 0.40 u. Such a low mass suggests this particle is an alpha particle, typically consisting of 2 protons and 2 neutrons with a standard mass near 4 u, indicated here as a lightweight particle under relativistic effects.

Key Concepts

Ionization Measurements

Ionization measurements play a crucial role in identifying and analyzing particles in physics. When a charged particle moves through a medium, it can ionize atoms and molecules along its path. The extent of ionization can help determine the particle's properties. The specific measurement mentioned involves the determination of a "double charge" on a particle. This implies that the particle carries twice the elementary charge, which is the charge on a proton, given by \( e = 1.60 \times 10^{-19} \text{ C} \). In our scenario, the particle carries a charge \( q = 2e \).

• Double Charge: It tells us that the particle has more than just a basic proton's charge; it is twice that.
• Significance: Knowing the charge helps to make use of formulas linking charge to motion, like the radius of curvature.
• Tools: Ionization chambers are typically used to help identify particles based on their charge.

This information combined with measurements of speed and the radius of curvature in a magnetic field gives us a method to calculate other vital properties, such as momentum and mass.

Relativistic Momentum

Momentum changes when particles move at speeds close to that of light, which is when relativistic effects come into play. The relativistic momentum \( p \) is not just the mass times velocity, as is usual for slower speeds. It incorporates the Lorentz factor \( \gamma \), reflecting how time and length alter at high velocities.To calculate this:

• Formula: \( p = \gamma mv \), where \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \).
• Calculation: First compute \( \gamma \) using the given speed \( v = 0.710c \).
• Application: Plugging \( \gamma \) into the momentum equation helps determine the true momentum at relativistic speeds.

Understanding this concept is important in modern physics, as particles in accelerators often reach relativistic speeds, affecting the calculated outcomes and ultimately influencing experimental interpretations.

Radius of Curvature

The radius of curvature is vital in understanding how charged particles move in magnetic fields. It provides insights into the relationship between the motion of particles, their momentum, and their charge.The main formula used is \( r = \frac{mv}{|q|B} \), where:

• \( r \): The radius of curvature of the particle's path.
• \( v \): The speed of the particle.
• |q|: The absolute value of the charge of the particle.
• \( B \): The magnetic field strength.

By rearranging this equation to solve for \( p \), you gain the particle's momentum \( p = |q|Br \). This value is essential in tackling more advanced problems involving particle dynamics in magnetic fields. Knowing how a particle's path curves gives direct information about its momentum when navigating these fields.

Alpha Particle

An alpha particle is a type of ionizing particle consisting of 2 protons and 2 neutrons, effectively making it a helium nucleus \( ^4_2\text{He} \). In particle physics, identifying such particles involves assessing aspects like their charge and mass.

• Charge: They carry a positive charge of \( +2e \) because of the two protons.
• Mass: Standard is around 4 u (atomic mass units), although relativistic effects in high-speed particles can appear to lower this.
• Detection: Often encountered in radiation studies, where ionization trails can indicate their presence.

In the essence of the given problem, the low calculated mass when interpreted via relativistic effects, helps identify the particle as an alpha particle due to its known properties. Such identification confirms consistency with ionization measurement theories in particle physics.