Problem 53 - Physics

Question

A capacitor has a capacitance of \(2.5 \times 10^{-8} \mathrm{~F}\). In the charging process, electrons are removed from one plate and placed on the other plate. When the potential difference between the plates is \(450 \mathrm{~V}\), how many electrons have been transferred?

Step-by-Step Solution

Answer

Approximately \( 7.03 \times 10^{13} \) electrons have been transferred.

Step 1: Understand the Relation Between Charge and Capacitance

The relationship between the charge \( Q \) on the plates of a capacitor, the capacitance \( C \), and the potential difference \( V \) is given by the formula \( Q = C \times V \). We need to calculate \( Q \), which represents the charge movement due to electron transfer.

Step 2: Calculate the Charge (Q)

Substitute the given capacitance \( C = 2.5 \times 10^{-8} \mathrm{~F} \) and the potential difference \( V = 450 \mathrm{~V} \) into the equation \( Q = C \times V \):\[ Q = (2.5 \times 10^{-8} \mathrm{~F}) \times (450 \mathrm{~V}) = 1.125 \times 10^{-5} \mathrm{~C} \].This value of \( Q \) represents the total charge transferred due to electrons.

Step 3: Convert Charge to Number of Electrons

The charge of one electron is approximately \( 1.6 \times 10^{-19} \mathrm{~C} \). To find the number of electrons \( n \) moved, use the formula \( n = \frac{Q}{e} \), where \( e \) is the elementary charge:\[ n = \frac{1.125 \times 10^{-5} \mathrm{~C}}{1.6 \times 10^{-19} \mathrm{~C/electron}} \approx 7.03125 \times 10^{13} \].

Key Concepts

Charge

In electronics, charge represents the quantity of electricity held or transported by a object or just a single particle. In the context of a capacitor, charge is the result of either transferring electrons to or from its plates. A capacitor stores this charge for various uses in electrical circuits. It does so by utilizing two conductive plates and accumulating charge on each plate during the charging process.The total amount of charge stored in a capacitor, denoted by the symbol \( Q \), is determined by its capacitance \( C \) and the potential difference \( V \) applied across its plates. The relationship is represented by the formula: \( Q = C \times V \).

• Capacitance \( C \) refers to the capacity of the capacitor to store charge. It's determined by factors such as plate size, plate separation, and dielectric material between the plates.
• The potential difference \( V \) refers to the voltage applied, which influences how much charge the capacitor will store.

No charge flow happens without potential difference, and the greater the potential difference, the more charge can be moved. The charge stored in a capacitor can be utilized in discharge processes to perform work, like powering electronic components momentarily.

Potential Difference

Potential difference, often referred to as voltage, is key in driving charge movement and storage in capacitors. It expresses the amount of work needed to move a charge between two points in an electric field.
The voltage applied across the plates of a capacitor incites electron flow, which moves electrons from one plate to another, eventually storing energy.

• Higher potential differences result in a greater amount of charge stored in a capacitor. This occurs because the electric field between the plates enhances as voltage rises, thus attracting more electrons to move across the field.
• Potential difference is measured in volts (\( V \)), and it dictates how energetically charges are pushed across the plates.

Potential difference plays a vital role in calculating the charge in capacitors, as seen in the provided formula \( Q = C \times V \). Understanding its role helps predict and control how capacitors function within electrical circuits.

Electron Transfer

Electron transfer is the essential mechanism by which capacitors store energy. When a voltage is applied, electrons are removed from one plate and transferred to the opposite plate. This movement creates a disparity in electron concentration, which stores energy as an electric field between the plates.
The amount of electron transfer is quantified by the charge moved, denoted by \( Q \) in the equation \( Q = C \times V \).

• The electron charge is a fundamental unit, with each electron carrying a charge of approximately \( 1.6 \times 10^{-19} \; \text{C} \).
• The number of electrons transferred is calculated by dividing the total charge \( Q \) by the charge of a single electron \( e \), using the formula \( n = \frac{Q}{e} \).

This aspect of a capacitor's charging process highlights the microscopic interactions essential for electrical operation, translating theoretical concepts into real-world applications by measuring the basic units of electron flow.