Question

A \(10.0-\mathrm{V}\) battery is connected to two parallel metal plates placed in a vacuum. An electron is accelerated from rest from the negative plate toward the positive plate. a) What kinetic energy does the electron have just as it reaches the positive plate? b) What is the speed of the electron just as it reaches the positive plate?

Short answer

Solution: The kinetic energy of the electron when it reaches the positive plate is 1.6 × 10^-18 J, and its speed is approximately 5.93 × 10^5 m/s.

Step-by-step solution

Calculate the work done on the electron

First, let's find the work done by the electric field on the electron. The work done on a charged particle across a potential difference is given by the following equation: \(W = qV\) where \(W\) is the work done, \(q\) is the charge of the electron, and \(V\) is the potential difference. We know the charge of the electron is \(e = -1.6 \times 10^{-19} \; \mathrm{C}\) and the voltage is \(10.0 \; \mathrm{V}\). Plug these values into the equation: \(W = (-1.6 \times 10^{-19} \; \mathrm{C})(10.0 \; \mathrm{V})\) \(W = -1.6 \times 10^{-18} \; \mathrm{J}\)

Find the kinetic energy using the work-energy theorem

The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, since the electron starts from rest, its initial kinetic energy is \(0\) and the change in kinetic energy will be the final kinetic energy. So, \(\Delta KE = KE_f - KE_i = KE_f - 0 = KE_f\) Thus, the final kinetic energy of the electron is equal to the work done on it by the electric field: \(KE_f = -1.6 \times 10^{-18} \; \mathrm{J}\) Note that the kinetic energy is negative because the electron is attracted to the positive plate, which makes the work done negative. The electron's kinetic energy is equal to the magnitude of the work done: \(KE_f = 1.6 \times 10^{-18} \; \mathrm{J}\)

Calculate the speed of the electron using the kinetic energy

The kinetic energy of an object with mass \(m\) and speed \(v\) is given by the equation: \(KE = \dfrac{1}{2} mv^2\) We want to find the speed \(v\) of the electron, so we will rearrange the formula to solve for \(v\): \(v = \sqrt{\dfrac{2KE}{m}}\) We know the mass of an electron is \(m = 9.11 \times 10^{-31} \; \mathrm{kg}\) and the kinetic energy we found in the previous step is \(KE = 1.6 \times 10^{-18} \; \mathrm{J}\). Plug these values into the equation: \(v = \sqrt{\dfrac{2(1.6 \times 10^{-18} \; \mathrm{J})}{9.11 \times 10^{-31} \; \mathrm{kg}}}\) \(v \approx 5.93 \times 10^5 \; \mathrm{m/s}\) So, the speed of the electron just as it reaches the positive plate is approximately \(5.93 \times 10^5 \; \mathrm{m/s}\).

Key concepts

Work Done by Electric Field

The idea of the work done by an electric field is central to understanding how forces within the field can influence charged particles like electrons. When a charged particle, such as an electron, moves through an electric field, the field does work on it. The work done on the particle is calculated by the equation:

\( W = qV \)

where \( W \) represents the work done, \( q \) is the charge of the particle, and \( V \) is the electric potential difference the particle travels through. For an electron accelerated between two metal plates by a potential difference, such as the scenario mentioned in our textbook problem, the electric field does work equal to the charge of the electron multiplied by the voltage difference between the plates. This work alters the kinetic energy of the electron as it moves from one plate to another.

In the case of our exercise, the negative value of work signifies that the direction of work is opposite to the direction of the electric field because the electron (negatively charged) is naturally attracted toward the positive plate. It’s important to note that the magnitude of the work reflects the energy change, regardless of the sign.

Work-Energy Theorem

The work-energy theorem is a fundamental concept in physics, providing a relationship between the work done on an object and its kinetic energy. It can be succinctly expressed as:

\( W = \triangle KE \)

Here, \( W \) is the work done on the object and \( \triangle KE \) symbolizes the change in kinetic energy of the object. According to the theorem, the total work done by all forces acting on an object is equal to the change in its kinetic energy.

Application to Electricity

When an electron is accelerated in an electric field, like in the given exercise, the work done by the electric field on the electron converts to kinetic energy. Remarkably, if the particle starts from rest, like the aforementioned electron, the entire work done by the field translates directly into kinetic energy, since the initial kinetic energy was zero. This principle allows us to calculate the resultant kinetic energy of the electron as it reaches the opposing metal plate.

Electric Potential and Charge Relationship

Electric potential, often represented as voltage, and charge are two foundational pillars within the realm of electromagnetism, tied together by a crucial relationship. The potential difference (or voltage) between two points in an electric field reflects the potential energy difference per unit charge. In simpler terms, it can be thought of as the energy that would be imparted to a charged particle should it be able to move from one point to another within the field. The formula:

\( V = \frac{W}{q} \)

connects the electric potential difference (\( V \)) with the work done (\( W \)) and the charge (\( q \)). By rearranging it, we identify that the work done by the field is the product of the charge and the potential difference.

• Higher voltage means more potential energy available for the charged particle per unit of charge.
• Greater charge means more work or energy transfer for the same potential difference.

In our exercise scenario, we see that an electron's kinetic energy and its final velocity upon reaching the positive plate can be derived from the potential difference across the plates and the charge of the electron. Thus, understanding electric potential and the charge relationship allows us to predict and quantify the kinetic outcomes for charged particles within an electric field.