Question

An electron is accelerated from rest through a potential of \(1.0 \cdot 10^{6} \mathrm{~V}\). What is its final speed?

Short answer

Answer: The final speed of the electron is approximately \(1.874 \cdot 10^{7} \mathrm{\frac{m}{s}}\).

Step-by-step solution

Identify given values and constants

In this problem, we have the following data: Potential, V = \(1.0 \cdot 10^{6} \mathrm{~V}\) (Voltage through which the electron is accelerated) Electron charge, e = \(-1.6 \cdot 10^{-19} \mathrm{~C}\) Electron mass, m = \(9.11 \cdot 10^{-31} \mathrm{~kg}\)

Calculate the work done on the electron

The work (W) done on the electron as it accelerates through the electric potential (V) is given by: \(W = e \times V\) Plug in the values for e and V: \(W = -1.6 \cdot 10^{-19} \mathrm{~C} \times 1.0 \cdot 10^{6} \mathrm{~V}\) \(W = -1.6 \cdot 10^{-13} \mathrm{~J}\)

Use the work-energy theorem to relate the work done to the final kinetic energy

According to the work-energy theorem, the work done on the electron equals the change in its kinetic energy: \(W = KE_{final} - KE_{initial}\) Since the electron starts from rest, its initial kinetic energy is 0. Thus, \(W = KE_{final}\)

Calculate the final kinetic energy of the electron

From step 2 and step 3, we have: \( KE_{final} = -1.6 \cdot 10^{-13} \mathrm{~J}\)

Calculate the final speed of the electron using the kinetic energy formula

The formula for kinetic energy is given by: \(KE = \frac{1}{2}mv^2\) Where m is the mass of the electron and v is its final speed. We can now solve for the final speed (v) by rearranging the formula: \(v = \sqrt{\frac{2 \times KE}{m}}\) Plug in the values for KE and m: \(v = \sqrt{\frac{2 \times -1.6 \cdot 10^{-13} \mathrm{~J}}{9.11 \cdot 10^{-31} \mathrm{~kg}}}\) \(v \approx 1.874 \cdot 10^{7} \mathrm{\frac{m}{s}}\) So, the final speed of the electron is approximately \(1.874 \cdot 10^{7} \mathrm{\frac{m}{s}}\).

Key concepts

Electric Potential

Electric potential, often referred to simply as voltage, is a fundamental concept in physics that describes the potential energy per unit charge at a point in an electric field. When dealing with the electricity, it's like considering the height of a hill on a hiking trail: the higher you go, the more potential energy you accumulate.
In a practical sense:

• Electric potential helps us understand how much work is needed to move a charge within an electric field.
• It is measured in volts (V), where one volt equals one joule per coulomb.
• When a charge moves through an electric potential difference, work is done on it, which can alter its speed.

In the case of an electron, moving through an electric potential difference means it is accelerated. The work done by the electric potential results in a change in its kinetic energy. Simply put, the electric field does work on the electron, converting potential energy into kinetic energy, which increases the electron's speed.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. For something very small and fast like an electron, there's a specialized formula to calculate this energy: \[ KE = \frac{1}{2}mv^2 \]where:

• \( KE \) is the kinetic energy
• \( m \) is the mass of the electron
• \( v \) is the velocity or speed of the electron

By knowing the mass and speed of the electron, you can determine its kinetic energy. It's important to remember:

• The faster an object moves, the more kinetic energy it has.
• Kinetic energy is always a positive value because it involves squared variables.

In scenarios like electron acceleration through a potential difference, the electron's initial kinetic energy is zero (it starts from rest). As work is done, the potential energy converts to kinetic energy, increasing its speed. This principle illustrates the direct transformation from stored energy due to position (potential) into the energy due to motion (kinetic).

Work-Energy Theorem

The work-energy theorem is a pivotal principle in classical mechanics, establishing a relationship between work done and changes in kinetic energy. It's formulated as:\[ W = \Delta KE \]where:

• \( W \) is the work done on the object
• \( \Delta KE \) is the change in kinetic energy

In simple terms:

• This theorem states that the work done on an object is equal to the change in its kinetic energy.
• If more work is done on the object, its speed (and thus kinetic energy) will increase.
• Conversely, if negative work is done (like applying brakes), the object slows down.

For an electron being accelerated by an electric potential, the work done on the electron by the electric field is transferred entirely to its kinetic energy because it starts from rest. As a result, this work can be directly equated to the final kinetic energy of the electron, effectively boosting its speed through the acceleration process. Thus, the work-energy theorem connects the level of work done and the resulting speed change in clear, relatable terms.