Problem 74 - Fundamentals of Physics

Question

A \(100 \mathrm{~W}\) lamp has a steady current of \(0.83 \mathrm{~A}\) in its filament. How long is required for \(1 \mathrm{~mol}\) of electrons to pass through the lamp?

Step-by-Step Solution

Answer

It takes approximately 32.22 hours for 1 mole of electrons to pass through the lamp.

Step 1: Understand the problem

We need to calculate the time it takes for 1 mole (6.022 x 10^23 electrons) to pass through a lamp with a current of 0.83 A. The current (A) tells us how many electrons pass through per second.

Step 2: Recall key formula

The key formula connecting current and charge is: \( I = \frac{Q}{t} \), where \(I\) is the current in amperes, \(Q\) is the charge in coulombs, and \(t\) is time in seconds. Our task is to solve for \(t\).

Step 3: Calculate total charge of 1 mole of electrons

Each electron has a charge of \(1.6 \times 10^{-19}\) coulombs. For one mole, the total charge \(Q =\) number of electrons \(\times\) charge per electron = \(6.022 \times 10^{23} \times 1.6 \times 10^{-19}\).

Step 4: Solve for time

Substitute the total charge \(Q\) and the current \(I = 0.83 \) A into the formula \( I = \frac{Q}{t} \) to solve for \(t\). So, \( t = \frac{Q}{I} \). Compute \( t = \frac{6.022 \times 10^{23} \times 1.6 \times 10^{-19}}{0.83} \).

Step 5: Calculate the value

Perform the arithmetic to get \( t = \frac{9.6352 \times 10^4}{0.83} \), which approximates \( t \approx 1.16 \times 10^5 \) seconds.

Step 6: Convert time to hours

To convert the time from seconds to hours, divide by 3600 (since there are 3600 seconds in an hour). Calculate \( t \approx \frac{1.16 \times 10^5}{3600} \approx 32.22 \) hours.

Key Concepts

Mole Concept

In chemistry and physics, the mole is a fundamental unit of measurement used to express the amount of a substance. One mole of any substance contains exactly \(6.022 \times 10^{23}\) entities, which could be atoms, molecules, ions, or electrons.
This number is known as Avogadro's number.
In our problem, we deal with one mole of electrons. This means we have \(6.022 \times 10^{23}\) electrons passing through the lamp. Understanding the mole concept is crucial when calculating large quantities, like the total number of electrons flowing through a circuit.

Electron Charge

Electrons carry a fundamental electric charge, denoted as \(e\). The charge of a single electron is \(1.6 \times 10^{-19}\) coulombs. This is a very small amount of charge, reflecting the tiny size and mass of an electron.
When many electrons are considered together, their total charge can be significant. For instance, one mole of electrons has a huge total charge:

• Total Charge (\(Q\)) = Number of Electrons \( \times \) Charge of One Electron
• \( Q = 6.022 \times 10^{23} \times 1.6 \times 10^{-19}\)

Calculating this gives a substantial amount of charge passing with a steady current.

Time Conversion

Time conversion is often necessary when dealing with physical phenomena. In physics problems, time is commonly expressed in seconds, but more practical units like minutes or hours are often desired for human comprehension.
In this problem, after calculating the time required in seconds, we convert it into hours for easier understanding.

• To convert seconds to hours, divide the seconds by 3600, because there are 3600 seconds in an hour.
• Example from our solution: If \( t = 1.16 \times 10^5 \) seconds, then \( t \approx 32.22 \) hours.

Such conversions are essential for applying physics in real-world contexts.

Coulombs

Coulombs are the unit of electric charge in the International System of Units (SI). The symbol is \(C\). One coulomb equals the charge of approximately \(6.24 \times 10^{18}\) electrons.
Coulombs measure the quantity of electricity transported by 1 ampere in 1 second. This unit is essential in calculating how many electrons transfer in a circuit.

• In our problem, the total electric charge \(Q\), calculated as \(Q = 6.022 \times 10^{23} \times 1.6 \times 10^{-19}\), must be expressed in coulombs.
• This illustrates the relationship between the microscopic world of electrons and the macroscopic world of electronics.

Understanding coulombs allows us to bridge these scales in electric current problems.

Current Calculation

The concept of electric current is central to analyzing electrical circuits. Current, denoted as \(I\), is defined as the rate at which charge flows through a surface. It is measured in amperes (A), where 1 ampere is equivalent to 1 coulomb per second.
The fundamental formula relating current \(I\) to charge \(Q\) and time \(t\) is:

• \(I = \frac{Q}{t}\)

From this, if \(Q\) and \(I\) are known, we can solve for \(t\), the time:

• \(t = \frac{Q}{I}\)

In our example, we substitute the total charge of a mole of electrons and the given current to find the time for them to pass.
Understanding this calculation equips students to solve a variety of circuit-related problems effectively.