Question

An electron is accelerated from rest through a potential difference of \(370 \mathrm{~V}\). What is its final speed?

Short answer

Answer: The final speed of the electron is approximately \(8.64 \times 10^5\) m/s.

Step-by-step solution

Write down the known values

We are given: Potential difference, V = 370 V Electron charge, e = \(-1.6 \times 10^{-19} \mathrm{C}\) Electron rest mass, m = \(9.11 \times 10^{-31} \mathrm{kg}\) Initial speed, \(v_0 = 0\)

Find the work done by the electric field on the electron

As the electron is accelerated from rest by the electric field, work is done on the electron and gets converted into kinetic energy. We can calculate the work (W) done using the formula: \(W = eV\) Where: \(W\) is the work done, \(e\) is the electron charge, and \(V\) is the potential difference. Plug in the known values and calculate work done: \(W = (-1.6 \times 10^{-19} \mathrm{C})(370 \mathrm{V})\) \(W = -5.92 \times 10^{-17} \mathrm{J}\) Note that the negative sign indicates that the electron gains kinetic energy.

Calculate the final kinetic energy

Since the electron starts from rest, the work done by the electric field will be equal to the gained kinetic energy (K). The negative sign just means that the work is done against the electric field. So, \(K = |-5.92 \times 10^{-17} \mathrm{J}| = 5.92 \times 10^{-17} \mathrm{J}\)

Compute the final speed

Now, let's use the formula relating kinetic energy to the mass and speed of the electron: \(K = \frac{1}{2}mv^2\) We can rearrange terms to find the final speed of the electron (v): \(v = \sqrt{\frac{2K}{m}}\) Plug in the values for \(K\) and \(m\): \(v = \sqrt{\frac{2(5.92 \times 10^{-17} \mathrm{J})}{9.11 \times 10^{-31} \mathrm{kg}}}\) \(v \approx 8.64 \times 10^5 \mathrm{m/s}\) Therefore, the final speed of the electron is approximately \(8.64 \times 10^5\) m/s.

Key concepts

Electron Acceleration

When an electron accelerates, it's essentially speeding up or changing its velocity. This occurs due to the presence of an electric field, which exerts a force on the electron because it carries a charge. In our exercise, an electric field is created by a potential difference of 370 V, and an electron, initially at rest, is put in this field. This potential difference is like a 'hill' that the electron 'rolls down', gaining speed as it goes.

It's crucial to remember that acceleration is due to the potential difference and doesn't depend on the amount of charge an electron has, but on the ratio of electric force to its mass. In essence, the larger the potential difference, the greater the acceleration, leading to a higher final speed, provided the electron's mass remains constant. This ties deeply into understanding how potential differences can influence particle speeds in applications such as particle accelerators and cathode ray tubes.

Kinetic Energy Calculation

Understanding kinetic energy calculation is essential for solving problems involving moving particles, like electrons in our case. Kinetic energy (KE) is the energy an object has due to its motion, calculated with the formula \( KE = \frac{1}{2}mv^2 \), where 'm' is the mass and 'v' is the velocity of the object. For an electron, which is a particle of very small mass, the kinetic energy can become significant with high velocities.

In the exercise, we see that the work done by the electric field is entirely converted to the electron's kinetic energy. Since an electron is a subatomic particle, this energy might seem small with a value in joules, but on a quantum level, it's substantial enough to cause transitions in atomic and molecular systems or even damage biological tissues.

Work-Energy Principle

The work-energy principle is a fundamental concept in physics that states the work done by all forces acting on an object will result in a change in the object's kinetic energy. In the context of our problem with the electron being accelerated by an electric field, the principle tells us that the work done by the electric force, when moving the electron through a potential difference, is transferred into the kinetic energy of the electron.

Considering the work-energy principle helps us understand energy conversion processes. It's particularly important when dealing with charged particles in electric fields, as it simplifies the calculation of speed changes without needing to solve complex force-based equations. Furthermore, it's also the foundational principle behind many technologies, such as electric motors and generators, where electrical energy is converted into mechanical work or vice versa.