Question

In a cathode ray tube, electrons initially at rest are accelerated by a uniform electric field of magnitude $4.0 \times 10^{5} \mathrm{N} / \mathrm{C}\( during the first \)5.0 \mathrm{cm}$ of the tube's length; then they move at essentially constant velocity another \(45 \mathrm{cm}\) before hitting the screen. (a) Find the speed of the electrons when they hit the screen. (b) How long does it take them to travel the length of the tube?

Short answer

(a) The speed of the electrons when they hit the screen is the final speed after acceleration, given by the equation \(v^2 = 2ad\). To find the actual speed (v), take the square root of the result. (b) The total time it takes for the electrons to travel the length of the tube can be found by adding the time it takes to travel the accelerated distance (\(t_1\)) and the time it takes to travel the constant-velocity distance (\(t_2\)). This is given by the equation \(T = t_1 + t_2\).

Step-by-step solution

Identify given values

We are given the following values: - Electric field (E): \(4.0 \times 10^5 \mathrm{N} / \mathrm{C}\) - Distance traveled under acceleration (d₁): 5.0 cm - Distance traveled at a constant velocity (d₂): 45 cm - Electron charge (e): \(1.6 \times 10^{-19} \mathrm{C}\) - Electron mass (m): \(9.1 \times 10^{-31} \mathrm{kg}\) Note that we need to convert the distances into meters, so d₁ = 0.05 m and d₂ = 0.45 m.

Find the acceleration of the electrons

Using the electric field (E) and the charge of the electron (e), we can find the force experienced by the electrons: \(F = e \times E\) Now, we can find the acceleration (a) using the mass of the electron (m) and the force (F): \(a = F/m\)

Find the final speed of the electron after acceleration

Since the electrons are initially at rest, we can use the equation of motion for constant acceleration: \(v^2 = u^2 + 2ad\) As the initial velocity (u) is 0, this simplifies to: \(v^2 = 2ad\) This equation allows us to find the final speed of the electron (v) after being accelerated by the electric field.

Calculate the time it takes to travel the accelerated distance

Using the final speed (v) and the distance traveled during acceleration (d₁), we can find the time it takes for the electrons to travel this distance using the equation of motion for constant acceleration: \(v = u + at\) As the initial velocity (u) is 0, this simplifies to: \(t_1 = d_1/a\)

Calculate the time it takes to travel the constant-velocity distance

Next, we need to find the time it takes for the electrons to travel the remaining distance (d₂) at the constant velocity (v). Using the distance formula: \(d = vt\) We can find the time \(t_2\) it takes to travel this distance: \(t_2 = d_2/v\)

Find the total time and answer the question

To find the total time (T) it takes for the electron to travel the length of the tube, we add the time it takes to travel the accelerated distance (\(t_1\)) and the time it takes to travel the constant-velocity distance (\(t_2\)): \(T = t_1 + t_2\) Now we can answer the problem: (a) The speed of the electrons when they hit the screen is v. (b) The total time it takes for the electrons to travel the length of the tube is T.