Electron Force Calculation In Transformer Oil

Hey guys! Ever wondered about the forces at play between tiny electrons, especially when they're hanging out in a substance like transformer oil? Well, let's dive into a fascinating physics problem that explores just that. We're going to calculate the force between two electrons separated by a distance of 1 cm in transformer oil. This involves some fundamental physics principles, so let's break it down step by step. It is important to understand the principles that involve in the calculation of force, charge and medium.

Understanding the Basics

Before we jump into the calculations, let's make sure we're all on the same page with the key concepts involved. This will help you grasp the underlying physics and make the solution much clearer. We will cover Coulomb's Law, electron charge, permittivity, and the effect of a medium (like transformer oil) on the electric force.

Coulomb's Law: The Force Between Charges

At the heart of this problem is Coulomb's Law, a cornerstone of electrostatics. This law describes the electrostatic force between two charged particles. In simpler terms, it tells us how strongly charged objects attract or repel each other. The force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This means that larger charges exert a stronger force, and the force decreases rapidly as the distance between the charges increases. Mathematically, Coulomb's Law is expressed as:

F = k * |q1 * q2| / r^2

Where:

• F is the electrostatic force
• k is Coulomb's constant (approximately 8.9875 × 10^9 N⋅m^2/C2)
• q1 and q2 are the magnitudes of the charges
• r is the distance between the charges

The absolute value signs around q1 * q2 ensure that we're dealing with the magnitude of the force, as force is a vector quantity (it has both magnitude and direction). The direction of the force is along the line connecting the two charges; it's attractive if the charges have opposite signs and repulsive if they have the same sign.

Electron Charge: The Fundamental Unit

The electron is a fundamental subatomic particle with a negative electric charge. The magnitude of this charge is a fundamental constant of nature, denoted by 'e'. The value of the electron charge is approximately:

e = -1.602 × 10^-19 Coulombs (C)

This tiny charge might seem insignificant, but it's the building block of all electrical phenomena. In our problem, we're dealing with two electrons, each carrying this fundamental negative charge. The negative sign indicates that electrons are negatively charged, which is crucial for determining the direction of the force between them (they will repel each other, as like charges repel).

Permittivity: How Materials Affect Electric Fields

Permittivity is a measure of how much a material can resist the formation of an electric field within it. It's a property that describes how a material polarizes in response to an electric field, thereby reducing the field inside the material. A vacuum has a fundamental permittivity, denoted by ε₀ (epsilon naught), which is approximately:

ε₀ = 8.854 × 10^-12 Farads per meter (F/m)

When we introduce a material between charges, it affects the electric force between them. The material's ability to reduce the electric field is quantified by its relative permittivity (εr), also known as the dielectric constant. The absolute permittivity (ε) of a material is then given by:

ε = εr * ε₀

In our problem, we have transformer oil as the medium, which has a relative permittivity (εr) of 2.3. This means transformer oil reduces the electric field, and hence the force between the electrons, by a factor of 2.3 compared to a vacuum.

The Role of Transformer Oil

Transformer oil is an insulating oil commonly used in electrical transformers. Its primary function is to provide electrical insulation and cooling. Because it's an insulator, it has a higher permittivity than a vacuum. This higher permittivity plays a crucial role in reducing the electric field strength and, consequently, the force between charged particles within the oil. In our scenario, the transformer oil's permittivity will reduce the repulsive force between the two electrons compared to what it would be in a vacuum.

Problem Setup

Okay, now that we've covered the essential physics concepts, let's set up our problem. This involves identifying the given values, the unknowns, and the appropriate formulas to use. Breaking the problem down like this makes it much easier to solve.

Given Values

First, let's list all the information we have:

• Distance between the electrons (r): 1 cm = 0.01 meters (We need to convert to meters for consistency in units)
• Charge of an electron (e): -1.602 × 10^-19 C
• Permittivity of vacuum (ε₀): 8.854 × 10^-12 F/m
• Relative permittivity of transformer oil (εr): 2.3

These are the known quantities that we'll use in our calculations. It's crucial to have the correct units (SI units are preferred) to ensure accurate results.

Unknown Value

The quantity we need to find is the force (F) between the two electrons in the transformer oil. This is the target of our calculation.

Relevant Formula

We'll use Coulomb's Law, but we need to modify it to account for the presence of the transformer oil. The modified formula is:

F = |q1 * q2| / (4 * π * ε * r^2)

Where:

• F is the electrostatic force
• q1 and q2 are the magnitudes of the charges (both are the charge of an electron in our case)
• ε is the permittivity of the medium (transformer oil), which is ε = εr * ε₀
• r is the distance between the charges
• π is the mathematical constant pi (approximately 3.14159)

This formula is derived from Coulomb's Law by incorporating the effect of the medium's permittivity. The 4π factor comes from the geometry of the electric field around a point charge.

Step-by-Step Solution

Alright, let's get our hands dirty with the actual calculations! We'll follow a clear, step-by-step approach to ensure we arrive at the correct answer. Each step is crucial, so let's take our time and be precise.

Step 1: Calculate the Permittivity of Transformer Oil

First, we need to find the absolute permittivity (ε) of the transformer oil using the formula:

ε = εr * ε₀

Plugging in the values:

ε = 2.3 * 8.854 × 10^-12 F/m

ε ≈ 2.036 × 10^-11 F/m

So, the permittivity of transformer oil is approximately 2.036 × 10^-11 F/m. This value will be used in the Coulomb's Law equation to account for the oil's effect on the electric force.

Step 2: Apply Coulomb's Law

Now, we'll use the modified Coulomb's Law formula to calculate the force between the electrons:

F = |q1 * q2| / (4 * π * ε * r^2)

We know that q1 = q2 = -1.602 × 10^-19 C (the charge of an electron), ε ≈ 2.036 × 10^-11 F/m, and r = 0.01 m. Plugging these values into the formula:

F = |(-1.602 × 10^-19 C) * (-1.602 × 10^-19 C)| / (4 * π * 2.036 × 10^-11 F/m * (0.01 m)^2)

Step 3: Perform the Calculation

Let's break down the calculation:

• Numerator: |(-1.602 × 10^-19 C) * (-1.602 × 10^-19 C)| = 2.566 × 10^-38 C^2
• Denominator: 4 * π * 2.036 × 10^-11 F/m * (0.01 m)^2 ≈ 2.560 × 10^-14 N⋅m^2

Now, divide the numerator by the denominator:

F ≈ (2.566 × 10^-38 C^2) / (2.560 × 10^-14 N⋅m^2)

F ≈ 1.002 × 10^-24 N

Step 4: State the Result

Therefore, the force between the two electrons separated by 1 cm in transformer oil is approximately 1.002 × 10^-24 Newtons. This is an incredibly small force, which makes sense given the tiny charges and the relatively small distance.

Conclusion

So, there you have it! We've successfully calculated the force between two electrons in transformer oil. The key takeaway here is that Coulomb's Law, combined with an understanding of permittivity, allows us to quantify these fundamental interactions. The presence of the transformer oil significantly affects the force due to its permittivity, reducing the force compared to what it would be in a vacuum. This kind of calculation is not just an academic exercise; it's fundamental to understanding how electrical devices work and how materials interact with electric fields. Hope you found this breakdown helpful and insightful! Keep exploring the fascinating world of physics, guys! It is important to understand the concepts.