A Particle Has A Charge Of Q = 2e, Where E Is The Charge On An Electron. (a) Determine The Electric Potential
Understanding electric potential and charge interactions is fundamental in physics, especially in electrostatics. When dealing with particles carrying electric charge, calculating the electric potential they produce is crucial for analyzing various phenomena, from atomic interactions to large-scale electromagnetic systems. In this article, we focus on a specific scenario: a particle with a charge \( Q = 2e \), where \( e \) is the elementary charge (the charge of a single electron). Our goal is to determine the electric potential created by this particle under given conditions.
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Fundamentals of Electric Charge and Electric Potential
Before delving into the specific problem, it’s essential to understand some basic concepts.
What Is Electric Charge?
Electric charge is a property of matter that causes it to experience a force when placed in an electric field. The two types of electric charges are positive and negative. The elementary charge, denoted by \( e \), is approximately:- \( e = 1.602 \times 10^{-19} \) coulombs (C)
What Is Electric Potential?
Electric potential, often simply called potential, is the electric potential energy per unit charge at a point in space due to electric charges. It is a scalar quantity, measured in volts (V), where:- 1 volt = 1 joule/coulomb (J/C)
Electric Potential Due to a Point Charge
The electric potential \( V \) at a distance \( r \) from a point charge \( Q \) is given by Coulomb's law:\[
V = \frac{1}{4\pi \varepsilon_0} \times \frac{Q}{r}
\]
where:
- \( \varepsilon_0 \) is the vacuum permittivity, approximately \( 8.854 \times 10^{-12} \, \text{F/m} \)
- \( Q \) is the charge in coulombs
- \( r \) is the distance from the charge in meters
This formula assumes the point charge is isolated in free space, and the potential is measured relative to a point at infinity where the potential is zero.
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Given Data and Problem Statement
The problem specifies:
- The particle's charge: \( Q = 2e \)
- \( e \): elementary charge \( \approx 1.602 \times 10^{-19} \, \text{C} \)
Our task:
- To determine the electric potential \( V \) at a specific point or distance, based on the information provided.
Note: Since the problem statement does not specify the distance \( r \), for comprehensive understanding, we will analyze the potential at an arbitrary distance \( r \) from the particle, and later, if needed, evaluate for specific distances.
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Step-by-Step Calculation of Electric Potential
1. Expressing the Charge \( Q \)
Since \( Q = 2e \):\[
Q = 2 \times 1.602 \times 10^{-19} \, \text{C} = 3.204 \times 10^{-19} \, \text{C}
\]
2. Applying the Electric Potential Formula
Using Coulomb's law:\[
V(r) = \frac{1}{4\pi \varepsilon_0} \times \frac{Q}{r}
\]
The constant \( \frac{1}{4\pi \varepsilon_0} \) is known as Coulomb's constant \( k \):
\[
k = 8.9875 \times 10^9 \, \text{Nm}^2/\text{C}^2
\]
Thus,
\[
V(r) = k \times \frac{Q}{r}
\]
Substituting the known values:
\[
V(r) = (8.9875 \times 10^9) \times \frac{3.204 \times 10^{-19}}{r}
\]
\[
V(r) = \frac{(8.9875 \times 10^9) \times 3.204 \times 10^{-19}}{r}
\]
Calculating numerator:
\[
8.9875 \times 10^9 \times 3.204 \times 10^{-19} \approx 2.880 \times 10^{-9}
\]
Therefore,
\[
V(r) = \frac{2.880 \times 10^{-9}}{r}
\]
where \( r \) is in meters, and \( V(r) \) will be in volts.
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Understanding the Significance of the Electric Potential
The potential \( V(r) \) reflects how the particle's charge influences the space around it. Key points include:
- Potential at different distances: As \( r \) increases, the potential decreases proportionally, approaching zero at infinity.
- Sign of potential: Since \( Q \) is positive (\( 2e \)), the potential is positive at all points, indicating a repulsive effect on other positive charges.
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Applications and Examples
To contextualize the calculation, here are some typical scenarios:
Example 1: Potential at 1 nm from the particle
Given \( r = 1\, \text{nm} = 1 \times 10^{-9} \, \text{m} \):\[
V(1\, \text{nm}) = \frac{2.880 \times 10^{-9}}{1 \times 10^{-9}} = 2.880\, \text{V}
\]
This implies that at 1 nanometer from the particle, the electric potential is approximately 2.88 volts.
Example 2: Potential at 1 μm from the particle
Given \( r = 1\, \mu \text{m} = 1 \times 10^{-6} \, \text{m} \):\[
V(1\, \mu \text{m}) = \frac{2.880 \times 10^{-9}}{1 \times 10^{-6}} = 2.88 \times 10^{-3} = 0.00288\, \text{V}
\]
The potential drops significantly with increasing distance.
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Considerations and Limitations
While the calculations above are straightforward for point charges in free space, several factors can influence real-world scenarios:
- Medium effects: The presence of materials other than vacuum, such as dielectric substances, alters the permittivity and thus the potential.
- Charge distribution: If the charge is not concentrated at a point but spread out, the potential calculation becomes more complex.
- Boundary conditions: Near conductive surfaces or in systems with specific boundary conditions, potential calculations need to incorporate additional factors.
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Summary and Conclusion
In this comprehensive analysis, we have:
- Clarified the concept of electric potential and its relation to point charges.
- Calculated the electric potential generated by a particle with charge \( Q = 2e \) at an arbitrary distance \( r \).
- Demonstrated how the potential diminishes with increasing distance.
- Provided practical examples to illustrate the potential at nanoscale and microscale distances.
Final formula:
\[
V(r) = \frac{2.880 \times 10^{-9}}{r} \quad \text{(volts, with \( r \) in meters)}
\]
This fundamental calculation forms the basis for more complex electrostatic problems involving multiple charges, distribution of charge, and potential energy considerations in physics and engineering.
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Additional Resources for Further Study
- Textbook: Introduction to Electrodynamics by David J. Griffiths
- Online simulations: Coulomb’s Law and Electric Potential calculators
- Educational videos: Electrostatics and Electric Potential explanations on platforms like Khan Academy
Remember: Precise calculation of electric potential is essential for designing electronic devices, understanding atomic interactions, and exploring electromagnetic phenomena across various scientific fields.
