Problem 56 - Fundamentals of Physics

Question

If a cat repeatedly rubs against your cotton slacks on a dry day, the charge transfer between the cat hair and the cotton can leave you with an excess charge of \(-2.00 \mu \mathrm{C}\). (a) How many electrons are transferred between you and the cat? You will gradually discharge via the floor, but if instead of waiting, you immediately reach toward a faucet, a painful spark can suddenly appear as your fingers near the faucet. (b) In that spark, do electrons flow from you to the faucet or vice versa? (c) Just before the spark appears, do you induce positive or negative charge in the faucet? (d) If, instead, the cat reaches a paw toward the faucet, which way do electrons flow in the resulting spark? (e) If you stroke a cat with a bare hand on a dry day, you should take care not to bring your fingers near the cat's nose or you will hurt it with a spark. Considering that cat hair is an insulator, explain how the spark can appear.

Step-by-Step Solution

Answer

(a) Approximately 1.25 x 10^13 electrons are transferred. (b) Electrons flow from you to the faucet. (c) You induce a positive charge in the faucet. (d) Electrons flow from the faucet to the cat. (e) The spark is due to charge build-up and redistributing upon contact.

Step 1: Understanding Charge Transfer

The charge transfer involves electrons moving. The excess charge of \( -2.00 \, \mu \mathrm{C} \) means we've gained electrons because electrons have a negative charge. To find the number of electrons transferred, we divide the total charge by the charge of a single electron. One electron carries a charge of \(-1.60 \times 10^{-19} \, \mathrm{C}\)."

Step 2: Calculating Number of Electrons

The formula to calculate the number of transferred electrons \( n \) is \( n = \frac{Q}{e} \), where \( Q = -2.00 \, \mu \mathrm{C} = -2.00 \times 10^{-6} \, \mathrm{C}\) and \( e = -1.60 \times 10^{-19} \, \mathrm{C} \). Substituting in the values, we get \[ n = \frac{-2.00 \times 10^{-6} \, \mathrm{C}}{-1.60 \times 10^{-19} \, \mathrm{C}} \approx 1.25 \times 10^{13} \text{ electrons} \] flowing to you.

Step 3: Direction of Electron Flow in Spark

Since you have excess electrons, these will flow from you to the faucet to discharge the negative charge.

Step 4: Induced Charge in the Faucet

Before the spark, your negative charge induces a positive charge in the faucet by repelling electrons away from the side closest to you, leaving an excess of positive charge there.

Step 5: Electron Flow in Spark With Cat

If the cat reaches toward the faucet, with the paw having fewer electrons, the electrons will flow from the faucet to the cat.

Step 6: Spark Formation with Insulating Cat Hair

Despite cat hair being an insulator, rubbing charges your hand, creating a potential difference. Approaching conductive materials (like the cat's nose) redistributes charges at the surface, causing a spark as the potential equalizes.

Key Concepts

Charge Transfer

Charge transfer is an interesting phenomenon that involves the movement of electric charge, specifically electrons. In the scenario where a cat rubs against cotton slacks, the friction causes a transfer of charges. The cat's fur and the clothing have different tendencies to attract electrons, causing electrons to move.

• If you end up with a charge of \(-2.00 \, \mu \mathrm{C}\), it indicates you have gained electrons.
• The cat hair donates some of its electrons to you through the rubbing action.

The charge of an electron is very small, \(-1.60 \times 10^{-19} \, \mathrm{C}\), so it takes a large number of electrons to reach the charge difference observed here.

Electron Flow

Electrons have a natural propensity to move from a region of high electron density (more negative charge) to lower electron density (or more positively charged regions). This movement, or flow, of electrons occurs to equalize the charge distribution:

• When rubbing a cat creates an excess of electrons on your body, touching a metal faucet lets those electrons flow to the faucet.
• The electron flow happens because metal is a good conductor, providing a path for electrons to travel.

This flow from you to the faucet results in a small shock, often noticeable as a spark.

Induced Charge

Induced charge is a concept where a charged object can cause a redistribution of charges in a nearby object without direct contact. Let's look at how it works:

• When you have an extra negative charge, and your hand approaches but does not touch an uncharged faucet, the electrons in the faucet are repelled.
• This leaves a positively charged area near the location closest to your fingers, known as induced positive charge.
• The charges in the faucet rearrange to minimize repulsion, which demonstrates induction.

Induced charge is a key principle in electrostatics and is critical to understanding how sparks form between objects.

Static Electricity

Static electricity results from charge imbalance that usually forms through contact and separation of materials, like the cat and the clothing. Though often unnoticed, it plays a crucial role in everyday experiences.

• An insulator such as cat fur doesn't easily allow electrons to move through it, storing charge as static electricity.
• When objects with different static charges come close, electrons may suddenly jump to balance the charge difference.
• This jump appears as a spark, which can be felt painfully and observed visually as a small flash.

Static electricity underpins many electrostatic occurrences and why seemingly trivial acts lead to noticeable effects such as shocks.