Problem 7 - Fundamentals of Physics

Question

A particle of charge \(1.8 \mu \mathrm{C}\) is at the center of a Gaussian cube \(55 \mathrm{~cm}\) on edge. What is the net electric flux through the surface?

Step-by-Step Solution

Answer

The net electric flux is approximately \(2.03 \times 10^{5} \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}\).

Step 1: Understand the Problem

We need to find the net electric flux through a Gaussian cube when a charged particle is placed at its center. The charge of the particle is given as \(1.8 \mu \mathrm{C}\) and the side length of the cube is \(55 \mathrm{~cm}\).

Step 2: Apply Gauss's Law

Gauss's Law states that the electric flux \(\Phi_E\) through any closed surface is equal to the enclosed charge \(Q_{enc}\) divided by the permittivity of free space \(\varepsilon_0\). Mathematically, it is expressed as: \[ \Phi_E = \frac{Q_{enc}}{\varepsilon_0} \]

Step 3: Convert Charge to Coulombs

Convert the charge from microcoulombs to coulombs. The particle has a charge of \(1.8 \mu \mathrm{C} = 1.8 \times 10^{-6} \mathrm{C}\).

Step 4: Calculate Electric Flux

Substitute the charge \(Q_{enc} = 1.8 \times 10^{-6} \mathrm{C}\) and \(\varepsilon_0 = 8.854 \times 10^{-12} \mathrm{C}^2/(\mathrm{N} \cdot \mathrm{m}^2)\) into Gauss's Law equation: \[ \Phi_E = \frac{1.8 \times 10^{-6}}{8.854 \times 10^{-12}} \approx 2.03 \times 10^{5} \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C} \]

Step 5: Interpret the Result

The electric flux through the surface depends only on the enclosed charge and not on the size or shape of the surface, hence the cube dimension is not needed for this calculation.

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that helps us understand how electric fields interact with matter. It relates the electric flux passing through a closed surface to the charge enclosed by that surface. This law is mathematically expressed as: \[ \Phi_E = \frac{Q_{enc}}{\varepsilon_0} \]where \(\Phi_E\) is the electric flux, \(Q_{enc}\) is the enclosed charge, and \(\varepsilon_0\) is the permittivity of free space. Gauss's Law tells us that the electric flux through a closed surface depends solely on the net charge inside that surface. This is a powerful concept because it simplifies complex electric field calculations by focusing on the symmetry and distribution of charges rather than the field details.Important points to remember about Gauss's Law:

• It applies to closed surfaces, also known as Gaussian surfaces.
• The shape of the surface doesn't affect the flux, only the charge inside does.
• It's particularly useful when dealing with symmetric charge distributions.

Understanding Gauss's Law allows us to solve problems like calculating the electric flux through surfaces enclosing charges, making it a valuable tool in physics.

Electric Charge

Electric charge is a fundamental property of matter that gives rise to electric fields and forces. Charges can be positive or negative, and they interact with each other according to the inverse square law. In our exercise, we have a charge of \(1.8 \mu \mathrm{C}\), which can be converted to \(1.8 \times 10^{-6} \mathrm{C}\) for standard units.Key points about electric charge:

• Like charges repel, while opposite charges attract.
• The unit of electric charge is the coulomb (C).
• Charge is quantized, meaning it exists in discrete amounts, typically in multiples of the elementary charge \(e\), approximately \(1.602 \times 10^{-19} \mathrm{C}\).

Electric charge plays a crucial role in defining the behavior of electric fields, influencing how forces are exerted on other charged objects in the vicinity. In Gauss's Law, the charge enclosed by the Gaussian surface is the determining factor for calculating the electric flux.

Permittivity of Free Space

Permittivity of free space, denoted by \(\varepsilon_0\), is a fundamental constant that describes how electric fields interact with a vacuum or free space. Its value is approximately \(8.854 \times 10^{-12} \mathrm{C}^2/(\mathrm{N} \cdot \mathrm{m}^2)\).Understanding the role of permittivity:

• It appears in Gauss's Law, connecting charge to electric flux.
• Permittivity of free space helps determine the strength of electric forces between charges in a vacuum.
• In mediums other than vacuum, permittivity is higher, as materials affect how electric fields propagate.

In our problem, permittivity of free space is used to calculate the electric flux through the Gaussian cube enclosing the charged particle. While permittivity may seem abstract, it is crucial in accurately describing electric phenomena in physics.