Problem 43 - Fundamentals of Physics

Question

How much work must be done to increasc the speed of an electron (a) from \(0.18 c\) to \(0.19 \mathrm{c}\) and (b) from \(0.98 \mathrm{c}\) to \(0.99 \mathrm{c}\) ? Note that the speed increase is \(0.01 \mathrm{c}\) in both cascs.

Step-by-Step Solution

Answer

For part (a), work done is \(0.002m_0c^2\); for part (b), it is \(2.06m_0c^2\).

Step 1: Understand the Problem

The problem is asking for the work done on an electron to increase its speed. This requires using the concept of relativistic kinetic energy, as the speeds are close to the speed of light ( c ).

Step 2: Recall the Relativistic Kinetic Energy Formula

The relativistic kinetic energy is given by: \[ KE = (\gamma - 1)m_0 c^2 \]where \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \) is the Lorentz factor, \( m_0 \) is the rest mass of the electron, and \( v \) is the velocity of the electron.

Step 3: Determine the Work Done

The work done is the change in kinetic energy, which is:\[ \Delta KE = KE_{\text{final}} - KE_{\text{initial}} \]Substitute the relativistic energy expressions for the initial and final conditions.

Step 4: Calculate for Part (a)

For an increase from \(0.18c\) to \(0.19c\):1. Calculate \(\gamma\) for \(v_1 = 0.18c\) and \(v_2 = 0.19c\):\[ \gamma_1 = \frac{1}{\sqrt{1 - (0.18)^2}} \approx 1.016 \]\[ \gamma_2 = \frac{1}{\sqrt{1 - (0.19)^2}} \approx 1.018 \]2. Calculate \( \Delta KE \):\[ \Delta KE = (\gamma_2 - \gamma_1)m_0c^2 \approx (1.018 - 1.016)m_0c^2 \approx 0.002m_0c^2 \]

Step 5: Calculate for Part (b)

For an increase from \(0.98c\) to \(0.99c\):1. Calculate \(\gamma\) for \(v_1 = 0.98c\) and \(v_2 = 0.99c\):\[ \gamma_1 = \frac{1}{\sqrt{1 - (0.98)^2}} \approx 5.03 \]\[ \gamma_2 = \frac{1}{\sqrt{1 - (0.99)^2}} \approx 7.09 \]2. Calculate \( \Delta KE \):\[ \Delta KE = (\gamma_2 - \gamma_1)m_0c^2 \approx (7.09 - 5.03)m_0c^2 \approx 2.06m_0c^2 \]

Step 6: Interpret the Results

The work done is different in both cases due to the relativistic effects becoming more pronounced at higher speeds. In part (a), the work done is relatively small, while in part (b), it is significantly larger.

Key Concepts

Work Done on Electron

When talking about the work done on an electron, it's important to understand that this work refers to the energy required to change the kinetic energy of the electron as it accelerates to a new velocity. This change in energy is influenced by the speed of the electron, especially as it approaches the speed of light. In physics, this energy calculation is different at very high speeds compared to everyday speeds and that's where the term 'relativistic' comes into play.

Typically, work is calculated as the product of force and the distance moved by the force. For an electron experiencing a change in speed, we use the relativistic kinetic energy formula.

• First, we determine the initial and final kinetic energies using the Lorentz factor.
• Then, the work done is the difference between these two energy states.

For instance, in the problem provided, to increase an electron's speed from \(0.18c\) to \(0.19c\), a moderate amount of work is required. In contrast, increasing from \(0.98c\) to \(0.99c\) demands significantly more work because the electron is nearing the speed of light.

Lorentz Factor

The Lorentz factor, \(\gamma\), is a measure used in relativistic physics to account for effects that increase as objects approach the speed of light. It essentially adjusts measurements like time, length, and mass according to the speed of the object in relation to light speed.

• Mathematically, it is expressed as \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\).
• This factor approaches infinity as velocity, \(v\), gets closer to the speed of light, \(c\).

In the context of the exercise, \(\gamma\) was calculated for two velocity scenarios.

• For a slower speed increase from \(0.18c\) to \(0.19c\), \(\gamma\) changes minimally, indicating a small difference in required energy.
• However, for a high-speed increase from \(0.98c\) to \(0.99c\), \(\gamma\) changes drastically, highlighting the much greater amount of energy needed.

The Lorentz factor is fundamental to understanding relativistic physics because it indicates how drastic the effects are as objects move faster.

Relativistic Effects

Relativistic effects refer to the physical phenomena that occur at speeds approaching the speed of light, which are explained by Einstein's theory of relativity. These effects include changes in mass, time, and dimensions, all of which impact the energy calculations for high-speed particles like electrons.

• One key effect is time dilation, where time appears to slow down for objects moving close to the speed of light.
• Another is length contraction, which causes moving objects to appear shorter along the direction of motion.

In the exercise, the differences in work done across two velocity changes are examples of relativistic effects. At lower speeds, where velocities like \(0.18c\) and \(0.19c\) are involved, these effects are minor and manageable.
However, the scenario changes substantially at higher speeds, such as \(0.98c\) to \(0.99c\). Here, relativistic effects drastically increase the work needed to achieve even a small change in velocity. This is because the mass of an electron effectively increases, requiring more energy to accelerate further.
Understanding these effects is crucial in fields like particle physics and cosmology, where particles often travel at such high speeds.