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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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Most popular questions from this chapter

The explosive yield of the atomic bomb dropped on Hiroshima near the end of World War II was approximately 15.0 kilotons of TNT. One kiloton is about \(4.18 \cdot 10^{12} \mathrm{~J}\) of energy. Find the amount of mass that was converted into energy in this bomb.

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