The hot filament of the electron gun in a cathode ray tube releases electrons with nearly zero kinetic energy. The electrons are next accelerated under a potential difference of \(5.00 \mathrm{kV}\), before being steered toward the phosphor on the screen of the tube. a) Calculate the kinetic energy acquired by the electron under this accelerating potential difference. b) Is the electron moving at relativistic speed? c) What is the electron's total energy and momentum? (Give both values, relativistic and nonrelativistic, for both quantities.)

To calculate the kinetic energy of the electron, we need to use the expression for kinetic energy in terms of electric potential: \(K_e = e \cdot V\), where \(K_e\) is the kinetic energy, \(e\) is the elementary charge (\(1.60 \times 10^{-19} \mathrm{C}\)), and \(V\) is the potential difference (\(5000 \mathrm{V}\)). Therefore, \(K_e = (1.60 \times 10^{-19}) \cdot (5000)\) \(K_e = 8.00 \times 10^{-16}\,\mathrm{J}\).

To determine if the electron is moving at relativistic speeds, we first find the electron's velocity using the non-relativistic formula for kinetic energy: \(K_e = \frac{1}{2}m_ev^2\), where \(v\) is the velocity of the electron, \(m_e\) is the electron's mass (\(9.11 \times 10^{-31}\,\mathrm{kg}\)). Then we can find, \(v^2 = \frac{2K_e}{m_e}\). \(v = \sqrt{\frac{2(8.00 \times 10^{-16})}{(9.11 \times 10^{-31})}}\) \(v ≈ 1.33 \times 10^{7} \mathrm{m/s}\). We can determine if this speed is relativistic by comparing it to the speed of light (\(c = 3 \times 10^{8} \mathrm{m/s}\)). In our case, \(\frac{v}{c} ≈ 0.044\). Since this value is significantly less than 1, the electron is not moving at a relativistic speed.

Relativistic: Total energy: \(E = \frac{m_ec^2}{\sqrt{1-\left(\frac{v}{c}\right)^2}}\) Momentum: \(p = \frac{m_ev}{\sqrt{1-\left(\frac{v}{c}\right)^2}}\) Non-relativistic: Total energy: \(E = K_e\) Momentum: \(p = m_ev\) Using the values calculated in Step 1 and Step 2, we have the following values for the electron's total energy and momentum: Relativistic: - Total energy: \(E \approx 8.19 \times 10^{-14}\,\mathrm{J}\) - Momentum: \(p \approx 2.87 \times 10^{-23}\,\mathrm{kg\,m/s}\) Non-relativistic: - Total energy: \(E = 8.00 \times 10^{-16}\,\mathrm{J}\) - Momentum: \(p \approx 2.82 \times 10^{-23}\,\mathrm{kg\, m/s}\)