Question

If an electron moves from rest from a point at which potential is 50 volt, to another point at which potential is 70 volt, then its kinetic energy in the final state will be \(\ldots .\) (A) \(1 \mathrm{~N}\) (B) \(3.2 \times 10^{-18} \mathrm{~J}\) (C) \(3.2 \times 10^{-10} \mathrm{~J}\) (D) 1 dyne

Short answer

The short answer based on the provided step-by-step solution is: The final kinetic energy of the electron after moving from an initial potential of 50 V to a final potential of 70 V is \(3.2 \times 10^{-18} \mathrm{~J}\) (Option B).

Step-by-step solution

Understand the concepts of potential and kinetic energy

Potential energy is the energy an object possesses because of its position in a force field or that a system has stored because of the relative positions of its components. In the case of an electron, its potential energy is due to its position in an electric field. Kinetic energy is the energy an object possesses because of its motion. In this exercise, we will find the change in kinetic energy as the electron moves from one potential to another.

Find the change in potential energy

The electron moves from an initial potential of 50 V to a final potential of 70 V. The change in potential energy, ∆U, can be found by the equation: ∆U = q * ∆V Where q is the charge of an electron (q = -1.6 × 10^(-19) C), and ∆V is the change in potential, which in this case is (70 V - 50 V).

Calculate the change in potential energy

Calculate the change in potential energy using the equation from step 2: ∆U = (-1.6 × 10^(-19) C) * (70 V - 50 V) ∆U = -1.6 × 10^(-19) C * (20 V) ∆U = -3.2 × 10^(-18) J Since the electron is moving from a lower potential (50 V) to higher potential (70 V), it's losing potential energy, thus the negative sign.

Determine the change in kinetic energy

By the conservation of energy principle, the change in potential energy is equal to the change in kinetic energy: ∆K = -∆U ∆K = 3.2 × 10^(-18) J

Identify the correct answer

Now we can see that the final kinetic energy of the electron after moving from 50 V to 70 V is 3.2 × 10^(-18) J. Comparing this value with the given options, we find that the correct answer is: (B) \(3.2 \times 10^{-18} \mathrm{~J}\)

Key concepts

Potential Energy

Potential energy is a fundamental concept in physics. It refers to the energy stored in an object due to its position or arrangement. In this specific context, an electron experiences potential energy because of its position in an electric field. When the electron is at a higher potential (like moving from 50V to 70V), it stores more potential energy.

The potential energy difference between two points in an electric field can be calculated using the formula:

• \( \Delta U = q \cdot \Delta V \)

where \(\Delta U\) is the change in potential energy, \(q\) is the charge of the electron, and \(\Delta V\) is the change in voltage between two points. Understanding potential energy helps us figure out how energy is transformed from one form to another as particles, like electrons, move within fields.

Electric Field

An electric field surrounds any electric charge and exerts a force on other charges in the field, either attracting or repelling them. It is a crucial aspect of our understanding of electric forces and energy. Electric fields are represented by vectors that point from positive to negative charges. The strength of an electric field is measured by how much force it exerts on a charge placed within it, defined as:

• Electric field (\(E\)) = Force (\(F\)) / Charge (\(q\))

When an electron moves within an electric field, its potential energy changes due to the field's influence. As the electron transitions from one point to another with a different electric potential, its position within the field is altered, leading to changes in potential and kinetic energy.

Conservation of Energy

The conservation of energy principle is a key concept in physics, stating that energy cannot be created or destroyed; it can only change forms. This principle holds true in an electric field where potential energy and kinetic energy of an electron can exchange forms.

When an electron moves from a lower potential to a higher potential, it loses potential energy, which is converted into kinetic energy. The formula illustrating conservation of energy in this scenario is:

• \( \Delta K = -\Delta U \),

where \(\Delta K\) is the change in kinetic energy and \(\Delta U\) is the change in potential energy. By applying this principle, it helps in calculating the kinetic energy gained by an electron as it moves through an electric field, ensuring that the overall energy remains constant.

Charge of Electron

The charge of an electron is a fundamental constant in physics, crucial for understanding interactions in electric and magnetic fields. Each electron carries a negative charge of \(-1.6 \times 10^{-19}\) Coulombs. This small charge may seem insignificant, but it plays a massive role in electric phenomena and energy calculations. When explaining changes in energy as an electron moves through an electric field, this charge becomes essential.

In energy calculations, the charge determines how much work is done by or on the electron as it moves between different potentials. For example, as seen in our exercise, the change in potential energy calculations used the electron's charge as a multiplier to determine the resulting energy change. Therefore, recognizing and utilizing the charge of an electron is vital to understanding and predicting the behavior of electrons under various electrical conditions.