Problem 17 - Fundamentals of Physics

Question

A uniformly charged conducting sphere of \(1.2 \mathrm{~m}\) diameter has surface charge density \(8.1 \mu \mathrm{C} / \mathrm{m}^{2}\). Find (a) the net charge on the sphere and (b) the total electric flux leaving the surface.

Step-by-Step Solution

Answer

(a) The net charge is approximately \(3.66 \times 10^{-5} \text{ C}\). (b) The total electric flux is approximately \(4.14 \times 10^{6} \text{ N}\cdot\text{m}^2/\text{C}\).

Step 1: Find the Radius of the Sphere

The diameter of the sphere is given as \(1.2 \text{ m}\). To find the radius \(r\), divide the diameter by 2: \(r = \frac{1.2}{2} = 0.6 \text{ m}\).

Step 2: Calculate the Surface Area of the Sphere

The formula for the surface area of a sphere is \(A = 4\pi r^2\). Substitute \(r = 0.6\text{ m}\) into the formula: \[A = 4\pi (0.6)^2 = 4\pi \times 0.36 \approx 4.52 \text{ m}^2\].

Step 3: Calculate the Net Charge on the Sphere

Surface charge density \(\sigma\) is given as \(8.1 \mu\text{C/m}^2\). The net charge \(Q\) on the sphere is \(Q = \sigma \times A\). Substitute \(\sigma = 8.1 \times 10^{-6} \text{ C/m}^2\) and \(A = 4.52 \text{ m}^2\) into the formula: \[Q = 8.1 \times 10^{-6} \times 4.52 \approx 3.66 \times 10^{-5} \text{ C}\].

Step 4: Calculate the Total Electric Flux

According to Gauss's law, the total electric flux \(\Phi\) through the surface of a sphere is \(\frac{Q}{\varepsilon_0}\), where \(\varepsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{N}\cdot\text{m}^2\) is the permittivity of free space. Substitute \(Q = 3.66 \times 10^{-5} \text{ C}\) into the formula: \[\Phi = \frac{3.66 \times 10^{-5}}{8.85 \times 10^{-12}} \approx 4.14 \times 10^{6} \text{ N}\cdot\text{m}^2/\text{C}\].

Key Concepts

Electric Flux

Electric flux is a measure of the number of electric field lines passing through a given surface. It is an important concept in electromagnetism, especially when dealing with Gauss's law. Electric flux, denoted by \( \Phi \), is calculated by taking the dot product of the electric field \( \mathbf{E} \) and the area vector \( \mathbf{A} \). Mathematically, it is expressed as:

• \( \Phi = \mathbf{E} \cdot \mathbf{A} \)

For closed surfaces, Gauss's law relates the total electric flux to the charge enclosed by the surface. According to Gauss's law, the total electric flux is:

• \( \Phi = \frac{Q}{\varepsilon_0} \)

where \( Q \) is the total charge enclosed and \( \varepsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \text{ C}^2/\text{N}\cdot\text{m}^2 \). In the context of a conducting sphere, the charge resides on the surface, making Gauss's law especially applicable. Since the sphere is symmetrical, the electric flux calculation becomes straightforward. Just apply the formula directly using the net charge found on the sphere. This concise approach helps when analyzing electric fields around symmetrical objects.

Surface Charge Density

Surface charge density is a concept that describes how much electric charge is distributed over a surface area. It is denoted by \( \sigma \) and is defined as the charge \( Q \) per unit area \( A \) of a surface.

• \( \sigma = \frac{Q}{A} \)

For a sphere, this means that the charge is distributed evenly across the entire surface. Given in units such as microcoulombs per square meter (\( \mu\text{C/m}^2 \)), it provides a simple way to quantify charge in problems involving surface areas. In our exercise, the surface charge density of the sphere was provided, allowing us to calculate the net charge \( Q \) by multiplying \( \sigma \) by the total surface area \( A \).

• \( Q = \sigma \times A \)

This relationship is key in electrostatics, especially when considering objects like conductors where charges rearrange themselves to reside on the surface. This ensures that the electric field inside a conductor remains zero, maintaining equilibrium.

Conducting Sphere

A conducting sphere is a classic example used in electrical engineering and physics to simplify the study of electric fields and potential. In a conductor, charges are free to move, redistributing until they reach an equilibrium state. This results in all excess charge residing on the surface of the sphere. When a conducting sphere is isolated, the surface charge density is uniform. This uniform distribution leads to a very interesting property: the electric field inside a conducting sphere is zero. This is because charges arrange themselves to cancel out internal electric fields, creating a balance. Outside the sphere, the electric field behaves as if all the charge is concentrated at the center of the sphere. Therefore, the sphere can be treated like a point charge for calculations at external points. Similarly, when evaluating the total electric flux through the surface of such a sphere, Gauss's law becomes very powerful. It allows us to easily relate the charge to the flux, making analysis straightforward. This makes conducting spheres valuable in electrostatics for both conceptual understanding and practical applications.