Problem 13 - College Physics

Question

\(\bullet\) As you walk across a synthetic-fiber rug on a cold, dry win- ter day, you pick up an excess charge of \(-55 \mu \mathrm{C}\) . (a) How many excess electrons did you pick up? (b) What is the charge on the rug as a result of your walking across it?

Step-by-Step Solution

Answer

(a) Approximately 3.44 x 10^14 excess electrons. (b) The rug has a +55 µC charge.

Step 1: Identify given information

You have been given an excess charge of \(-55 \mu \mathrm{C}\) on your body after walking across the rug. This charge is negative, indicating an excess of electrons on your body.

Step 2: Convert microcoulombs to coulombs

Convert the charge from microcoulombs (\(\mu C\)) to coulombs (\(C\)) using the conversion 1 microcoulomb = \(10^{-6}\) coulombs: \(-55 \mu C = -55 \times 10^{-6} C = -5.5 \times 10^{-5} C\).

Step 3: Calculate the number of excess electrons

Use the elementary charge \(e = 1.6 \times 10^{-19} C\) to find the number of excess electrons \(n_e\) using the formula: \[n_e = \frac{|q|}{e}\] where \(q = -5.5 \times 10^{-5} C\). Calculate \(n_e = \frac{5.5 \times 10^{-5}}{1.6 \times 10^{-19}} \approx 3.44 \times 10^{14}\).

Step 4: Determine the charge on the rug

According to the conservation of charge, if your body gains an excess charge of \(-55 \mu C\) (electrons), the rug loses the same amount of charge. Hence, the rug has a positive charge of \(+55 \mu C\) as a result.

Key Concepts

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when near other electrically charged objects. It is a key concept in electrostatics, which studies how charged particles interact. Charges come in two types: positive and negative, typically associated with protons and electrons, respectively.

When two objects have the same type of charge, they repel each other, while opposite charges attract. This principle is fundamental in explaining numerous natural phenomena and the workings of various electronic devices.

For instance, in the original exercise, when you walk across a synthetic rug, your body picks up excess electrons and becomes negatively charged. This excess charge results in an imbalance, making electrostatic interactions with other objects possible.

In practical applications, understanding electric charge helps in designing circuits, making batteries, and even preventing electrostatic discharge in sensitive electronic components.

Elementary Charge

The elementary charge is the smallest unit of electric charge that is possible in ordinary matter, denoted by the symbol \(e\). It represents the charge of a single proton or an electron, with a magnitude of approximately \(1.6 \times 10^{-19}\) coulombs.

This value is crucial in calculating the number of excess electrons or protons in any given charge. For instance, in the exercise, the excess charge of \( -55 \mu \text{C} \) on your body was converted into a numeric count of electrons using this elementary charge value.

Whenever dealing with incidents involving static electricity—like shuffling across the carpet—this concept helps in quantifying the charges involved, which ultimately aids in practical calculations and safety measures in electrical setups.

Charge Conservation

Charge conservation is a fundamental principle in physics, stating that the total electric charge in an isolated system remains constant. In simpler terms, electric charge can neither be created nor destroyed; it only transfers from one object to another.

In the scenario provided, as you accumulate electrons and gain a negative charge, the rug loses an equal amount of electrons, gaining a positive charge of equal magnitude, precisely \(+55 \mu \text{C}\).

This principle of conservation ensures the balance of charge within any interaction, safeguarding some of the key ideas in both chemistry and physics. By understanding charge conservation, students can predict the outcome of various electrical interactions and uphold the fundamental laws of electromagnetism.