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Chapter 19: Problem 44

The energy that should be added to an electron, to reduce its de-Broglie wavelengths from \(10^{-10} \mathrm{~m}\) to \(0.5 \times 10^{-10} \mathrm{~m}\), will be (A) four times the initial energy. (B) thrice the initial energy. (C) equal to the initial energy. (D) twice the initial energy.

Short Answer

Expert verified
To find the energy that should be added to the electron, we use the de-Broglie wavelength formula, the momentum formula, and the kinetic energy formula. First, we compute the initial and final kinetic energy using the given wavelengths: \(K_{initial}\) from \(\lambda_{initial} = 10^{-10} \mathrm{~m}\) and \(K_{final}\) from \(\lambda_{final} = 0.5 \times 10^{-10} \mathrm{~m}\). Then, we find the energy difference, \(\Delta K = K_{final} - K_{initial}\). Finally, we compare the added energy with the initial energy to select the correct option.

Step by step solution

01

Write down the de-Broglie wavelength formula

The de-Broglie wavelength formula is given by: \[ \lambda = \frac{h}{p} \] where \(\lambda\) is the de-Broglie wavelength, \(h\) is the Planck's constant (\(h = 6.626 \times 10^{-34} \text{ Js}\)), and \(p\) is the momentum of the particle.
02

Write down the formula for momentum and kinetic energy

The momentum formula is given by: \[ p = \sqrt{2\cdot m_e \cdot K} \] where \(p\) is the momentum, \(m_e\) is the electron's mass (\(m_e = 9.109\times 10^{-31}\, \text{kg}\)), and \(K\) is the kinetic energy of the electron.
03

Find the initial kinetic energy using the initial wavelength

We first find the initial momentum using the initial de-Broglie wavelength: \[ p_{initial} = \frac{h}{\lambda_{initial}} \] Now, we can find the initial kinetic energy (K_i) using the momentum formula: \[ K_{initial} = \frac{p_{initial}^2}{2\cdot m_e} \]
04

Find the final kinetic energy using the final wavelength

Similarly, we can find the final momentum using the final de-Broglie wavelength: \[ p_{final} = \frac{h}{\lambda_{final}} \] Then, we can find the final kinetic energy (K_f) using the momentum formula: \[ K_{final} = \frac{p_{final}^2}{2\cdot m_e} \]
05

Calculate the energy that should be added to the electron

The energy that should be added to the electron is given by the difference between the final kinetic energy and the initial kinetic energy: \[ \Delta K = K_{final} - K_{initial} \]
06

Compare the added energy with the initial energy

Now, we compare the energy that should be added to the electron with the initial energy to find which one of the options is correct. For example, if the added energy is equal to twice the initial energy, then the correct option would be (D) twice the initial energy.

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