Problem 59 - College Physics

Question

A vacuum tube has a vertical height of \(50.0 \mathrm{~cm}\). An electron leaves from the top at a speed of \(3.2 \times 10^{6} \mathrm{~m} / \mathrm{s}\) downward and is subjected to a "typical" Earth field of \(150 \mathrm{~V} / \mathrm{m}\) downward. (a) Use energy methods to determine whether it reaches the bottom surface of the tube. (b) If it does, with what speed does it hit? If not, how close does it come to the bottom surface? (c) How does the gravitational force on the electron compare to the electric force on it, both in magnitude and in direction?

Step-by-Step Solution

Answer

Electron doesn't reach the bottom; it comes 0.194 m close. Electric force dominates gravity.

Step 1: Analyze the Forces on the Electron

The electron is subjected to two forces: gravitational force and electric force. The gravitational force is negligible compared to the electric force because the mass of the electron is extremely small. The electric force is given by \( F = eE \), where \( e \) is the charge of the electron (\( 1.6 \times 10^{-19} \) C) and \( E = 150 \) V/m.

Step 2: Determine the Work Done by Electric Force

The work done on the electron as it moves a distance \( d = 50.0 \) cm (or \( 0.5 \) m) is \( W = Fd = eEd \). Substituting \( e = 1.6 \times 10^{-19} \) C, \( E = 150 \) V/m, and \( d = 0.5 \) m gives \( W = (1.6 \times 10^{-19})(150)(0.5) = 1.2 \times 10^{-17} \) J.

Step 3: Calculate the Initial Kinetic Energy of the Electron

The initial kinetic energy \( KE_i \) of the electron is \( KE_i = \frac{1}{2}mv_i^2 \), where \( m \) is the mass of the electron \( (9.11 \times 10^{-31} \) kg), and \( v_i = 3.2 \times 10^6 \) m/s. Calculating gives \( KE_i = \frac{1}{2}(9.11 \times 10^{-31})(3.2 \times 10^6)^2 \approx 4.66 \times 10^{-18} \) J.

Step 4: Determine If Electron Reaches the Bottom

Compare the work done by the electric force \( (1.2 \times 10^{-17} \) J) with the initial kinetic energy \( (4.66 \times 10^{-18} \) J). The work done by the electric field is greater than the initial kinetic energy of the electron, indicating that the electron does not have enough energy to reach the bottom of the tube.

Step 5: Find the Closest Approach to the Bottom

Since the electron cannot reach the bottom, we set the final kinetic energy to zero and use energy conservation: \( KE_i = W \). Solving \( \frac{1}{2}mv_i^2 = eEd' \) for \( d' \), \( d' = \frac{KE_i}{eE} = \frac{4.66 \times 10^{-18}}{1.6 \times 10^{-19} \times 150} \approx 0.194 \) m. The electron comes within \( 0.194 \) m of the bottom.

Step 6: Compare Gravitational and Electric Forces

The gravitational force on the electron \( F_g = mg = 9.11 \times 10^{-31} \times 9.81 \approx 8.93 \times 10^{-30} \) N. The electric force \( F_e = eE = 1.6 \times 10^{-19} \times 150 = 2.4 \times 10^{-17} \) N. The electric force is much greater than the gravitational force and acts in the same direction as the electron's motion (downward).

Key Concepts

Electron Motion in Electric Fields

When an electron moves through an electric field, it experiences a force due to the interaction between its charge and the field. Electrons, being negatively charged, will naturally be attracted towards positively charged regions or pushed away from negatively charged regions within an electric field. In our scenario, the electron is subjected to a downward electric field of 150 V/m. This field applies a force on the electron, which influences its motion. The force due to the electric field is calculated using the formula:

• Electric Force, \( F = eE \)
• where \( e \) is the charge of the electron (\( 1.6 \times 10^{-19} \) C)
• \( E \) is the electric field strength (150 V/m)

This force causes the electron to accelerate in the direction of the field. Understanding the motion of electrons in electric fields is crucial as it forms the basis of numerous technologies, such as cathode-ray tubes and particle accelerators.

Kinetic Energy Calculations

The kinetic energy of an electron is a measure of its motion energy, and it can be calculated using the kinetic energy formula:

• \( KE = \frac{1}{2}mv^2 \)
• where \( m \) is the mass of the electron (\( 9.11 \times 10^{-31} \) kg)
• \( v \) is the velocity of the electron

Initially, the electron has a velocity of \( 3.2 \times 10^6 \) m/s. By calculating its initial kinetic energy, we can understand how much energy the electron has to overcome forces acting against its motion. In this example, the initial kinetic energy turns out to be \( 4.66 \times 10^{-18} \) J. This energy is critical to determine whether the electron has enough energy to reach the bottom of the vacuum tube when subjected to opposing forces like the electric field.

Work-Energy Principle

The work-energy principle ties together the concepts of work and kinetic energy. It states that the work done on an object is equal to the change in its kinetic energy. This principle can be expressed as:

• \( W = \Delta KE = KE_f - KE_i \)

In practical scenarios, like our electron in the vacuum tube, when the electron moves under an electric field, work is performed by the electric force. The work done (\( W = eEd \)) affects the electron's kinetic energy as it moves. In our case, since the electron does not reach the bottom, the work done by the electric field is greater than its initial kinetic energy. Hence, all its initial kinetic energy is converted to work against the electric field before it can reach the bottom, resulting in a final kinetic energy that comes effectively to zero at its closest point.

Gravitational vs Electric Forces

To understand the motion of electrons, it's important to compare gravitational and electric forces acting on them. Although we often think of gravitational force as significant in our daily experience, for particles like electrons, the electric force usually dominates:

• Gravitational Force, \( F_{g} = mg \)
• Electric Force, \( F_{e} = eE \)
• Typically, \( F_{e} \gg F_{g} \) for electrons

In this specific exercise, the gravitational force on the electron is approximately \( 8.93 \times 10^{-30} \) N, while the electric force is \( 2.4 \times 10^{-17} \) N. The electric force is several orders of magnitude larger than the gravitational force, making it the primary force influencing the electron's motion in the tube. Both forces act downwards, but the sheer magnitude of the electric force renders the gravitational force negligible in determining the electron's trajectory.