Problem 5 - College Physics

Question

A rubber rod rubbed with fur acquires a charge of \(-4.8 \times 10^{-9} \mathrm{C} .\) (a) Is the charge on the fur (1) positive, (2) zero, or (3) negative? Why? (b) What is the charge on the fur, and how much mass is transferred to or from the rod? (c) How much mass has the rubber rod lost or gained?

Step-by-Step Solution

Answer

(a) Positive (b) Charge: +4.8 x 10^{-9} C, Mass transferred: 2.73 x 10^{-20} kg (c) Rubber rod gained 2.73 x 10^{-20} kg.

Step 1: Determine Charge on Fur

When a rubber rod is rubbed with fur, electrons are transferred from the fur to the rod, causing the rod to become negatively charged. Since the rod is negatively charged \((-4.8 \times 10^{-9} \mathrm{C})\), it means the fur must have lost electrons, resulting in a positive charge. Therefore, the charge on the fur is positive.

Step 2: Calculate Charge on Fur

By conservation of charge, the total charge of the system must remain zero. Therefore, the charge gained by the rod is equal in magnitude and opposite in sign to the charge lost by the fur. Hence, the charge on the fur is \(4.8 \times 10^{-9} \mathrm{C}\) (positive).

Step 3: Determine Number of Electrons Transferred

Using the charge of one electron, which is \(-1.6 \times 10^{-19} \mathrm{C}\), calculate the number of electrons transferred. The number of electrons \(n\) is given by \(n = \frac{-4.8 \times 10^{-9}}{-1.6 \times 10^{-19}} = 3 \times 10^{10}\).

Step 4: Calculate Mass Transferred

The mass of one electron is approximately \(9.11 \times 10^{-31} \mathrm{kg}\). Calculate the total mass transferred using the number of electrons: \(m = 3 \times 10^{10} \times 9.11 \times 10^{-31} = 2.73 \times 10^{-20} \mathrm{kg}\).

Step 5: Determine Mass Change of Rubber Rod

Since the rod gains electrons, the mass of the rod increases by the same mass that transferred from the fur, i.e., \(2.73 \times 10^{-20} \mathrm{kg}\).

Key Concepts

Conservation of Charge

When we talk about the conservation of charge, we mean that in an isolated system, the total amount of electric charge remains constant. This principle is fundamental in physics. In our scenario with the rubber rod and fur, the principle plays out clearly. When electrons transfer from the fur to the rod, the fur loses some negative charge, and the rod gains that exact amount of negative charge. Hence, the sum of charges between the rod and fur remains zero, illustrating charge conservation.

• An isolated system keeps its total charge constant.
• Electron transfer results in a balance between charge loss and gain.

Conservation of charge reassures us that the universe is neat and orderly. No charge magically appears or vanishes; it only moves around between objects.

Electron Transfer

Electron transfer is the process we observe when electrons move from one object to another, leading to a change in charge. In the case of a rubber rod rubbed with fur, electrons are transferred from the fur to the rubber rod. Electrons carry a negative charge, so transferring electrons from the fur to the rod leaves the fur with a positive charge and the rod with a negative charge.

Here's a quick breakdown:

• The fur loses electrons ➡️ becomes positively charged.
• The rod gains electrons ➡️ becomes negatively charged.

This process is essential in understanding not just this exercise, but also many real-world phenomena, including static electricity.

Mass Conservation

While charge conservation dominates our focus in electrostatics, mass conservation is equally important. Here, mass conservation means that when electrons move from the fur to the rod, mass is also transferred, although it's a minute amount. Each electron carries a tiny mass of approximately \(9.11 \times 10^{-31} \text{ kg}\). When billions of electrons transfer, the mass added to the rubber rod is equivalent to the total mass of electrons.

• The mass of one electron is \(9.11 \times 10^{-31} \text{ kg}\).
• Mass change occurs because electrons have mass.

The rod thus seems insignificantly heavier due to these extra electrons, but in practice, this added mass is usually too small to be detected with everyday tools. Nonetheless, mass conservation holds true, preserving total mass even when only charged particles are in play.