Problem 66 - Fundamentals of Physics

Question

A charged particle causes an electric flux of \(-750 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}\) to pass through a spherical Gaussian surface of \(10.0 \mathrm{~cm}\) radius centered on the charge. (a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the charge of the particle?

Step-by-Step Solution

Answer

(a) Flux remains the same: \(-750 \text{ N} \cdot \text{m}^{2}/\text{C}\). (b) Charge is \(-6.64 \times 10^{-9} \text{ C}\).

Step 1: Understand the Exercise

The exercise involves an electric charge and a Gaussian surface. Given an electric flux \(-750 \text{ N} \cdot \text{m}^{2} / \text{C}\) through a spherical Gaussian surface of radius \(10.0 \text{ cm}\), we need to find (a) the flux if the radius is doubled, and (b) the charge of the particle.

Step 2: Apply Gauss's Law

Gauss's Law states that the electric flux \(\Phi_E\) through a closed surface is equal to the charge \(Q\) enclosed divided by the permittivity of free space \(\epsilon_0\):\[\Phi_E = \frac{Q}{\epsilon_0}\]The given flux is \(-750 \text{ N} \cdot \text{m}^{2} / \text{C}\), meaning it depends only on the charge \(Q\), not the radius of the Gaussian surface.

Step 3: Solve Part (a) - Doubling the Radius

Since the flux through the Gaussian surface depends only on the charge enclosed and not the size of the Gaussian surface, the flux remains \(-750 \text{ N} \cdot \text{m}^{2} / \text{C}\) even if the radius is doubled.

Step 4: Calculate the Charge - Part (b)

Using Gauss's Law:\[\Phi_E = \frac{Q}{\epsilon_0} \Rightarrow Q = \Phi_E \cdot \epsilon_0\]Given \(\Phi_E = -750 \text{ N} \cdot \text{m}^{2} / \text{C}\) and \(\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2 / \text{N} \cdot \text{m}^2\):\[Q = -750 \cdot 8.85 \times 10^{-12} = -6.6375 \times 10^{-9} \text{ C}\]

Step 5: Result Interpretation

From the calculations, the charge causing the flux is \(-6.64 \times 10^{-9} \text{ C}\) (rounded to three significant figures).

Key Concepts

Electric Flux

Electric flux is a fundamental concept in electromagnetism, represented by the Greek letter \( \Phi_E \). It describes the amount of electric field passing through a given surface. Imagine electric field lines as arrows representing the direction of the field. Electric flux measures how many of these lines penetrate through a surface.The formula for electric flux is given by:- \( \Phi_E = E \cdot A \cdot \cos \theta \) where, - \( E \) is the electric field strength, - \( A \) is the area through which the field lines pass, - \( \theta \) is the angle between the field lines and the normal (perpendicular line) to the surface.Key Points:- If the surface is perpendicular to the electric field lines (\( \theta = 0 \)), all lines pass through, maximizing flux.- If \( \theta = 90^\circ \), no field lines pass through, and the flux is zero.Understanding electric flux helps visualize how electric fields interact with surfaces.

Gaussian Surface

A Gaussian surface is an imaginary closed surface used in applying Gauss's Law. Its purpose is to simplify the calculation of electric fields and flux for symmetric charge distributions. Some features of a Gaussian surface: - It must be closed, like a sphere or a cube. - The choice of shape depends on the symmetry of the problem, such as: - Spherical symmetry (use a sphere) - Cylindrical symmetry (use a cylinder) - It does not affect the actual physical setup or result; instead, it's a theoretical tool. When using Gauss's Law: - The surface should simplify the problem, allowing the use of symmetry to determine the electric field. In essence, Gaussian surfaces are conceptual tools that help us harness symmetry to make tough calculations simpler.

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electric field. It comes in two types: positive and negative. Like charges repel each other, while opposite charges attract each other.Important aspects of electric charge:- Quantized: Charge comes in integer multiples of the elementary charge (\( e \)).- Conserved: The total charge in a closed system remains constant.- Measured in Coulombs (C).In the context of Gauss's Law:- The electric flux through a closed surface is directly proportional to the charge enclosed within that surface.- More charge inside means more electric flux, irrespective of the surface's size.Understanding charge is crucial in predicting how objects will behave under the influence of electric forces.

Permittivity of Free Space

The permittivity of free space, denoted by \( \epsilon_0 \), is a constant that describes how electric fields interact with the vacuum of free space. It is a fundamental physical constant essential in the equations of electrostatics.Key properties:- Value: \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/ (\text{N} \cdot \text{m}^2) \)- Appears in formulas for electric force, capacitance, and famously, Gauss's Law: - \( \Phi_E = \frac{Q}{\epsilon_0} \)- It essentially regulates the strength of electric fields in a vacuum.By understanding \( \epsilon_0 \), we gain insights into how electric fields propagate through space, influencing calculations involving charges and fields. This constant helps in applying Gauss's Law accurately in theoretical scenarios and practical applications.