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Chapter 35: Problem 63

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?

Short Answer

Expert verified
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 by step solution

01

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 \(v_2 = 2.00 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\).
02

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 \(v_2 = 2.00 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\).
03

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.

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