Problem 3 - Physics

Question

A metal sphere has a charge of \(+8.0 \mu \mathrm{C}\). What is the net charge after \(6.0 \times 10^{13}\) electrons have been placed on it?

Step-by-Step Solution

Answer

The net charge of the sphere is \(-1.6 \mu \text{C}\).

Step 1: Understanding Charge Quantities

A single electron has a charge of approximately \(-1.6 \times 10^{-19} \text{ C}\). In this problem, there are \(6.0 \times 10^{13}\) electrons being added to the sphere's charge.

Step 2: Calculate Total Electron Charge

Calculate the total charge of the electrons by multiplying the number of electrons by the charge of one electron: \[q = (6.0 \times 10^{13}) \times (-1.6 \times 10^{-19} \text{ C}) = -9.6 \times 10^{-6} \text{ C}\] This is the total charge contributed by the added electrons.

Step 3: Determine the Net Charge of the Sphere

The initial charge of the sphere is \(+8.0 \times 10^{-6} \text{ C}\) (since \(\mu\text{C} = 10^{-6}\text{ C}\)). The net charge after the electrons are added is calculated by adding the initial charge to the total electron charge: \[\text{Net Charge} = 8.0 \times 10^{-6} \text{ C} + (-9.6 \times 10^{-6} \text{ C}) = -1.6 \times 10^{-6} \text{ C}\]

Step 4: Present the Net Charge Result

After the electrons are added, the sphere's net charge becomes \(-1.6 \times 10^{-6} \text{ C}\). The negative sign indicates that the sphere is now negatively charged.

Key Concepts

Charge of an electron

The concept of the charge of an electron is fundamental in understanding electrostatic phenomena. Electrons are subatomic particles that carry a negative charge, which is universally recognized to be approximately \(-1.6 \times 10^{-19} \; \text{Coulombs}\).- This negative sign indicates the type of charge, not a physical negative quantity. - It’s important to remember that an electron's charge is a constant value.
This means that no matter where an electron is found, it will always have this exact charge.
Given the extremely small nature of this charge, electrostatic calculations often involve large numbers of electrons.
Even though a single electron's charge is very small, the cumulative charge from a large number of electrons can lead to noticeable and significant effects on objects and their interactions.

Net charge calculation

Net charge calculation involves determining the total charge on an object after accounting for all positive and negative contributions. This is process crucial in countless physical applications.
The net charge is the overall charge of an object once all individual charges, such as from electrons, are summed.- To find the net charge in cases like the one in the exercise, first, calculate the total charge contributed by all added or removed charges.
- Simply multiply the number of particles by the charge of each particle. For example, adding electrons involves calculating the total negative charge by multiplying the number of electrons by the electron's charge.In the problem, you're adding \(6.0 \times 10^{13}\) electrons to a metal sphere that initially has a \(+8.0 \mu \text{C}\) charge.

First, calculate the charge from the electrons:\[n(-1.6 \times 10^{-19} \text{ C}) = -9.6 \times 10^{-6} \text{ C}\]Next, calculate the new total by adding the original sphere's charge:\[8.0 \times 10^{-6} \text{ C} + (-9.6 \times 10^{-6} \text{ C}) = -1.6 \times 10^{-6} \text{ C}\]This shows the sphere’s new net charge, indicating any shift due to the electron addition.

Metal sphere

In physics, a metal sphere commonly serves as a model for exploring electrostatic behaviors. Metals are excellent conductors of electricity, which means they allow free movement of charges along their surfaces.
This movement affects the distribution and accumulation of net charges on them. - When you add extra charges like electrons, the sphere’s initial charge distributes more broadly to balance out these new negative charges. A noteworthy property of a metal sphere is the phenomena of charge distribution. Any excess charge resides entirely on the surface, due to the repulsive forces between like charges—often referred to as Gauss's Law. - This makes spheres particularly interesting and wide-used in electrostatic experiments because changes in net charge can be directly measured and analyzed as a whole rather than needing to account for depth or layering. Consider a metal sphere carrying a net positive charge, like the one initially described in the exercise.
Adding a significant number of electrons switches the net charge from positive to negative.
This showcases how dynamic and responsive metal spheres are to changes in charge.