Problem 63 - University Physics with Modern Physics

Question

Using relativistic expressions, compare the momentum of two electrons, one moving at \(2.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}\) and the other moving at \(2.00 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\). What is the percentage difference between classical momentum values and these values?

Step-by-Step Solution

Answer

Question: Calculate the relativistic and classical momentum values for two electrons that are moving at \(2.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}\) and \(2.00 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\), respectively. Also, determine the percentage difference between the relativistic and classical momentum values for each electron.

Step 1: Calculate Relativistic Momentum

First, let's recall the expression for relativistic momentum: $$ p = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}} $$ Here, \(p\) is the relativistic momentum, \(m\) is the mass of the electron, \(v\) is the velocity of the electron, and \(c\) is the speed of light (\(3 \times 10^8 \mathrm{~m}/\mathrm{s}\)). We will calculate the relativistic momentum for both electrons, one with \(v_1 = 2.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}\) and the other with \(v_2 = 2.00 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\).

Step 2: Calculate Classical Momentum

To calculate the classical momentum, we use the simple expression: $$ p = mv $$ Here, \(p\) is the classical momentum, \(m\) is the mass of the electron, and \(v\) is the velocity of the electron. We will calculate the classical momentum for both electrons, one with \(v_1 = 2.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}\) and the other with \(v_2 = 2.00 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\).

Step 3: Calculate Percentage Difference

With the calculated relativistic and classical momentum values, we will now determine the percentage difference for each electron using the formula: $$ \text{Percentage Difference} = \frac{|\text{Relativistic Momentum} - \text{Classical Momentum}|}{\text{Classical Momentum}} \times 100 $$ We will find the percentage difference for both electrons, comparing their relativistic and classical momentum values.

Key Concepts

Classical Momentum

Classical momentum is a foundational concept in physics, often described using the formula \( p = mv \), where \( p \) denotes momentum, \( m \) represents mass, and \( v \) is velocity. It is a straightforward calculation that defines how an object in motion carries motion depending on its mass and speed.
For example, in the case of electrons moving at different velocities, their classical momentum can be calculated simply by multiplying their mass by their speed. This simplicity is one of the biggest advantages of using the classical formula, especially when velocities are much lower than the speed of light.

• Easy to use for everyday speeds
• Directly proportional to both mass and velocity
• Not accurate when objects approach speeds close to the speed of light

This classic approach works well in most day-to-day applications but begins to falter when objects move at velocities approaching the speed of light, where relativistic effects become significant.

Percentage Difference

The percentage difference is a useful measure to understand how much the classical model differs from relativistic predictions. It's calculated using the formula:\[\text{Percentage Difference} = \frac{|\text{Relativistic Momentum} - \text{Classical Momentum}|}{\text{Classical Momentum}} \times 100\]This formula gives us a percentage which describes how far off the classical calculations are from what actually happens at high speeds.
When velocities are moderate, the difference remains small and possibly negligible. However, when the speed increases significantly, especially close to the speed of light, the discrepancy grows:

• Helpful to evaluate model accuracy
• Shows increasing deviation at high velocities
• Illustrates the limits of classical mechanics

By understanding these differences, we gain insight into why relativistic physics is needed to accurately describe high-velocity motion.

Speed of Light

The speed of light, denoted as \( c \), is one of the fundamental constants of nature, valued at approximately \( 3 \times 10^8 \text{ m/s} \). It is the ultimate speed limit in the universe, beyond which traditional physics laws undergo significant alterations.
Several key facts about the speed of light include:

• Nothing with mass can reach or exceed this speed
• Fundamental in equations of relativity
• Affects the mass, time, and length of moving objects

In the context of momentum, when an object's speed approaches that of light, its relativistic momentum diverges from classical predictions. At such velocities, the relativistic momentum formula \( p = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}} \) needs to be used, reflecting the complex interplay at near-light speeds.

Electron Velocity

Electron velocity is crucial in understanding momentum at both classical and relativistic scales. Electrons are tiny particles moving through space, and their momentum can be analyzed through their velocities.
The two major velocities considered in this problem are:

• High velocity: \( 2.00 \cdot 10^8 \text{ m/s} \)
• Low velocity: \( 2.00 \cdot 10^3 \text{ m/s} \)

These two velocities illustrate the need to use different momentum concepts. For high electron velocity, relativistic momentum becomes significant because it's a substantial fraction of the speed of light. Conversely, low electron velocity allows classical calculations, as relativistic effects are negligible.
Understanding electron velocity in various conditions lets us appreciate the transition between classical and modern physics, showcasing how fundamental particles behave under different forces and conditions.