Problem 4 - College Physics

Question

A glass rod rubbed with silk acquires a charge of \(+8.0 \times 10^{-10} \mathrm{C} .\) (a) Is the charge on the silk (1) positive, (2) zero, or (3) negative? Why? (b) What is the charge on the silk, and how many electrons have been transferred to the silk? (c) How much mass has the glass rod gained or lost?

Step-by-Step Solution

Answer

(a) The silk has a negative charge. (b) Charge on the silk is -8.0 × 10^{-10} C; 5.0 × 10^{9} electrons transferred. (c) Rod lost 4.56 × 10^{-21} kg of mass.

Step 1: Understanding the Principle of Charge Transfer

When a glass rod is rubbed with silk, electrons are transferred from the glass rod to the silk, leaving the glass rod positively charged and the silk negatively charged.

Step 2: Determining the Charge on the Silk

Since electrons are transferred from the glass rod to the silk, the silk will acquire a negative charge equal in magnitude to the positive charge on the rod. Hence, the charge on the silk is negative and equals \[-8.0 \times 10^{-10} \mathrm{C}.\]

Step 3: Calculating the Number of Electrons Transferred

To find the number of electrons transferred, use the formula: \[ q = n \cdot e, \]where \( q \) is the charge, \( n \) is the number of electrons, and \( e \) is the charge of an electron \( (1.60 \times 10^{-19} \mathrm{C}).\) Solving for \( n \), we get:\[ n = \frac{8.0 \times 10^{-10} \mathrm{C}}{1.60 \times 10^{-19} \mathrm{C/electron}} = 5.0 \times 10^{9} \text{ electrons}. \]

Step 4: Determining the Mass Change of the Rod

Since electrons have been removed from the rod, its mass decreases. The mass of a single electron is \(9.11 \times 10^{-31} \mathrm{kg}.\) Therefore, the mass lost is:\[ 5.0 \times 10^{9} \text{ electrons} \times 9.11 \times 10^{-31} \mathrm{kg/electron} \approx 4.56 \times 10^{-21} \mathrm{kg}. \]

Key Concepts

Charge Transfer

When we talk about charge transfer, we're looking at how electric charges move from one object to another. In the case of the glass rod and silk, this process happens when they are rubbed together. The rubbing action causes electrons to move. This is because electrons are much lighter and more mobile than protons, which stay in place within the nucleus of atoms.

In this example, electrons are transferred from the glass rod to the silk. As electrons leave the rod, it starts losing negative charge and hence becomes positively charged. Conversely, the silk gains these electrons and acquires a negative charge. This transfer of electrons from one material to another is a classic example of charge transfer, and it plays a crucial role in everyday electrostatic phenomena.

Electron Transfer

Electron transfer is integral to understanding charge interactions. Electrons are subatomic particles with a negative charge and are much smaller than protons or neutrons. This makes them quite agile in terms of movement between materials. The electron transfer between the glass rod and silk sets the stage for a fascinating interplay of charges.

By knowing the charge of an electron, which is a fundamental physical constant at approximately \(1.60 \times 10^{-19} \text{C} \), we can calculate the number of electrons transferred. In our scenario, the silk receives a charge of \(-8.0 \times 10^{-10} \text{C} \), owing to the electrons it gained. By dividing this total charge by the charge of an individual electron, we can discover how many electrons moved between the materials. Here, the number is an astonishingly large \(5.0 \times 10^{9} \), or in other words, five billion electrons! This is a great illustration of how numerous electrons can dramatically shift what seems like a small charge.

Mass Change of Rod

While charge transfer happens, another subtle change is taking place. This involves the mass of the glass rod. When electrons leave an object, that object loses a bit of mass. Although each electron is incredibly tiny, about \(9.11 \times 10^{-31} \mathrm{kg} \), the removal of a large number of electrons \((5.0 \times 10^{9})\) adds up to more noticeable mass change.

In this scenario, we calculate the total mass change by multiplying the mass of a single electron with the total number of electrons transferred. This results in the glass rod losing approximately \(4.56 \times 10^{-21} \mathrm{kg} \). While this mass change might seem minuscule, it's an excellent example of how minute changes at the atomic level can have measurable consequences in the macro world. It's intriguing to see how something as simple as electron movement can influence an object's mass!