Question

A proton, initially at rest, is accelerated through a potential difference of \(500 .\) V. What is its final velocity?

Short answer

Answer: The final velocity of the proton is 4.38 * 10^5 m/s.

Step-by-step solution

Identifying given values

We are given the following values: - Potential difference, V = 500 V - Proton mass, m = 1.6726 * 10^{-27} kg - Initial velocity, v_0 = 0 (proton is initially at rest) - Charge of proton, q = 1.6022 * 10^{-19} C

Calculation of change in potential energy

We will first find the change in potential energy as the proton is accelerated through a potential difference of 500 V. We can calculate the change in potential energy using the formula: ∆PE = q * V where ∆PE is the change in potential energy, q is the charge of the proton, and V is the potential difference. Substituting the given values: ∆PE = (1.6022 * 10^{-19} C) * (500 V) = 8.011 * 10^{-17} J

Applying conservation of energy

As the proton is accelerated through the potential difference, its potential energy is converted into kinetic energy. We can use the conservation of energy principle: ∆PE = ∆KE where ∆KE is the change in kinetic energy. Since the proton is initially at rest, its initial kinetic energy is 0, and thus the change in kinetic energy is: ∆KE = KE_f - KE_0 = 8.011 * 10^{-17} J where KE_f is the final kinetic energy of the proton.

Using the kinetic energy formula

Now we can use the kinetic energy formula to relate the proton's final kinetic energy to its final velocity: KE_f = (1/2) * m * v_f^2 where m is the mass of the proton and v_f is the final velocity of the proton. Solving for v_f, we get: v_f = sqrt(2 * KE_f / m) Substituting the values for kinetic energy and mass: v_f = sqrt(2 * 8.011 * 10^{-17} J / (1.6726 * 10^{-27} kg)) = 4.38 * 10^5 m/s #Conclusion#: The final velocity of the proton after being accelerated through a potential difference of 500 V is 4.38 * 10^5 m/s.

Key concepts

Potential Difference

In the realm of physics, particularly in the study of electromagnetism, potential difference, often termed voltage, is a fundamental concept. It is the measure of the work needed to move a charge from one point to another against an electric field. In simpler terms, it's much like the pressure that pushes water through a pipe; similarly, potential difference drives the flow of electric charge in a circuit.

To visualize this, imagine two points in a space where electric charges are present. The potential difference between these two points is determined by the amount of energy required to move a unit charge from one point to the other. The standard unit for measuring potential difference is the volt (V). In the textbook exercise, a potential difference of 500 volts across the proton implies that energy is being supplied to the proton to move across this electric 'pressure'.

Moreover, as potential difference is the cause behind the acceleration of the proton, understanding its role is key for students to comprehend the process through which kinetic energy is gained by the particle.

Conservation of Energy

The principle of conservation of energy is a cornerstone in physics, asserting that within an isolated system, energy can neither be created nor destroyed; it can only change from one form to another. This concept has considerable implications in the field of mechanics, thermodynamics, and electrodynamics, among others.

In the context of proton acceleration, conservation of energy implies that the total energy before and after the acceleration process remains constant. Energy simply transforms from electric potential energy, which the proton holds due to its position in an electric field, into kinetic energy, which the proton exhibits as motion.

When a proton is accelerated by a potential difference, it does not 'gain' energy in the sense of increasing the total energy; instead, it converts the energy form. Understanding the conversion between these forms of energy is crucial when analyzing mechanical processes under the scope of physics.

Kinetic Energy

Kinetic energy represents the energy of motion. An object acquires kinetic energy when it is in motion, and the amount of kinetic energy depends on both the mass of the object and its velocity. Given by the mathematical expression \( KE = \frac{1}{2} mv^2 \), where m is mass and v is velocity, kinetic energy increases with the square of the velocity, making it a substantial factor in dynamic systems.

For a proton, initially at rest, that is subjected to a potential difference, the kinetic energy changes as the proton speeds up. Interestingly, even though the proton's mass remains constant, its velocity increases dramatically due to the acceleration caused by the electric field. This is why, after acceleration, the proton has significant kinetic energy, enough to be quantified and related to the potential difference that initially propelled it.

Electric Potential Energy

Electric potential energy is the energy a charged particle possesses by virtue of its position in an electric field. It is analogous to gravitational potential energy in mechanics, where an object's potential energy depends on its height above ground. In an electric field, potential energy is determined by the charge's position relative to other charges.

In our exercise, a proton experiences a change in potential energy when it is moved through a potential difference. This energy change is calculated by the formula \( \Delta PE = qV \), where \(q\) is the charge of the proton, and \(V\) is the potential difference. The concept is essential for students to understand how charged particles such as protons can store and transfer energy within electric fields and how this energy can subsequently be turned into kinetic energy to induce motion.