Question

How many excess protons are in a positively charged object with a charge of \(+398 \mathrm{mC}\) (millicoulombs)? The charge of 1 proton is \(+1.6 \times 10^{-19} \mathrm{C}\). (Hint: Use the charge of the proton in coulombs as a conversion factor between charge and number of protons.)

Short answer

There are approximately 2.49 billion excess protons in the object.

Step-by-step solution

Convert millicoulombs to coulombs

Since there are 1000 millicoulombs in one coulomb, convert the charge of the object from millicoulombs to coulombs by dividing by 1000. This will give us the charge in standard units (coulombs).

Find the number of protons

With the charge of one proton known, divide the total charge of the object by the charge of one proton to find the excess number of protons in the object.

Perform the calculation

Calculate the number of excess protons by dividing the object's charge (in coulombs) by the charge of a single proton and round the number to the closest whole number since you can't have a fraction of a proton.

Key concepts

Electric Charge

Electric charge is a fundamental property of particles that determines how they interact with electromagnetic fields and other charged particles. There are two types of electric charges: positive and negative. Protons carry a positive charge, while electrons carry a negative charge. The behavior of electric charges can be explained by the laws of electromagnetism, such as Coulomb's law, which states that like charges repel each other and opposite charges attract.

The unit of electric charge in the International System of Units (SI) is the coulomb (C). Electric charge is conserved in an isolated system, which means that the total charge remains constant over time. Understanding electric charge is essential for solving problems in electrostatics, circuit theory, and many areas of physics.

Coulombs to Millicoulombs Conversion

The coulomb is a very large unit of charge, and often electric charges are measured in smaller units, like millicoulombs (mC). To convert from millicoulombs to coulombs, you divide the number of millicoulombs by 1000, since 1 coulomb is equal to 1000 millicoulombs. This conversion is crucial when working with electric charges because it allows you to use standard units which simplifies calculations and ensures consistency.

To demonstrate, let's take the given charge of +398 mC. To convert this to coulombs, we simply divide 398 by 1000, obtaining +0.398 C. This step is foundational before proceeding to calculate quantities like the excess number of protons, which require charges to be expressed in the standard unit of coulombs.

Fundamental Charge of Proton

The fundamental charge of a proton is the elementary electric charge, which is a constant and one of the basic units in physics. It is denoted by the symbol 'e' and has a value of approximately +1.6 x 10^-19 coulombs. This represents the smallest unit of charge that is practical in physics and chemisty, since it is the charge per single proton (or the charge per single electron but with opposite sign).

In our context, to find the excess number of protons in the positively charged object, we use the fundamental charge of a proton as a conversion factor. By dividing the total charge of the object (already in coulombs) by the charge of a single proton, we can determine how many protons account for that excess charge. For instance, if an object has a charge of +0.398 C, we divide this by +1.6 x 10^-19 C/proton to find the total number of excess protons.