Question

A copper sphere of mass \(2 \mathrm{gm}\) contains about \(2 \times 10^{22}\) atoms. The charge on the nucleus of each atom is 29e. what fraction of electrons removed from the sphere to give it a charge of \(2 \mu \mathrm{c}\) ? (A) \(2 \times 10^{-10}\) (B) \(1.19 \times 10^{-12}\) (C) \(1.25 \times 10^{-11}\) (D) \(2.16 \times 10^{-11}\)

Short answer

The short answer is (B) \(1.19 \times 10^{-12}\).

Step-by-step solution

Calculate the total number of electrons in the sphere

The copper sphere contains about \(2 \times 10^{22}\) atoms, and the charge on the nucleus of each atom is \(29e\), where \(e\) is the elementary charge, \(1.6 \times 10^{-19} C\). Therefore, each copper atom has 29 electrons. To find the total number of electrons in the sphere, we need to multiply the number of atoms by the number of electrons per atom: \[ \text{Total electrons} = (\text{Number of atoms})(\text{Number of electrons per atom}) = (2 \times 10^{22})(29) \]

Calculate the number of electrons to be removed

Given that we need to give the sphere a charge of \(2\mu C\), we first need to find out how many electrons account for this charge. We can calculate this by dividing the desired charge by the charge of a single electron: \[ \text{Number of electrons to be removed} = \frac{\text{Desired charge}}{e} = \frac{2\mu C}{1.6 \times 10^{-19} C} = \frac{2 \times 10^{-6} C}{1.6 \times 10^{-19} C} \]

Calculate the fraction of electrons removed

Now that we have the total number of electrons and the number of electrons to be removed, we can find the fraction of electrons that need to be removed by dividing the number of electrons to be removed by the total number of electrons: \[ \text{Fraction of electrons removed} = \frac{\text{Number of electrons to be removed}}{\text{Total electrons}} = \frac{\frac{2 \times 10^{-6} C}{1.6 \times 10^{-19} C}}{(2 \times 10^{22})(29)} \]

Simplify and find the correct answer

Simplify the fraction and compare it with the provided options: \[ \text{Fraction of electrons removed} = \frac{2 \times 10^{-6}}{1.6 \times 10^{-19} \times 2 \times 10^{22} \times 29} = 1.19 \times 10^{-12} \] So, the answer is: (B) \(1.19 \times 10^{-12}\)

Key concepts

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electric field. The electric charge comes in two types, positive and negative.

• Positive charge: Carried by protons.
• Negative charge: Carried by electrons.

Charges exert forces on each other: like charges repel and unlike charges attract. This forces electrons to orbit the positively charged nucleus in atoms, maintaining electrical balance. A material can become charged, or experience an imbalance in charges, by gaining or losing electrons.
Understanding electric charge is essential for comprehending how materials interact electrically and plays a critical role in electrostatics, where charged objects influence one another without direct contact.

Elementary Charge

The elementary charge, represented by the symbol \( e \), is the smallest unit of electric charge that is possible in nature. It serves as a fundamental constant and is crucial in the study of electrostatics.

• The value of the elementary charge is \( 1.6 \times 10^{-19} \) coulombs.
• It is the charge carried by a single proton or the negative of that carried by a single electron.

The elementary charge is the basic unit of charge from which all charges are composed. For example, the charge on an ion is some multiple of the elementary charge. In our copper sphere example, since each copper atom has a nuclear charge of \( 29e \), the number of electrons equals 29 times the elementary charge to balance this nucleic charge under neutral conditions.

Fraction of Electrons

When discussing the removal of electrons from a neutral object, like in this exercise, it's useful to consider what fraction of the object's total electrons needs to be removed to achieve a given net charge. To find the fraction of electrons removed, you first need to know:

• The total number of electrons initially present in the object.
• The number of electrons corresponding to the desired net charge.

The fraction is calculated as the ratio of the number of electrons removed to the initial total: \[\text{Fraction} = \frac{\text{Desired Charge in Electrons}}{\text{Total Initial Electrons}}\]For the copper sphere, removing enough electrons to produce a \(2 \mu C\) charge represents an incredibly small fraction of the total electrons in the sphere, illustrative of how large a number of electrons are in even tiny pieces of everyday materials.

Copper Atoms

Copper atoms are essential in understanding the behavior of the copper sphere. Each copper atom comprises a nucleus with 29 protons and, typically, 29 electrons, balancing it out as neutral under normal conditions.Copper is a metal, and its electrons are integral to its properties:

• Copper's high conductivity derives from its electrons being easily shuffled from one atom to another.

In the given problem, we assessed the sphere containing about \(2 \times 10^{22}\) copper atoms. Because each has 29 electrons, calculating such vast numbers shows both the abundance of electrons and how a tiny fraction suffices to alter the sphere's overall charge significantly. Understanding copper atoms is significant in practical applications, such as in wires or electronic components, where their conductive properties become crucial.