Question

Another unit of charge is the electrostatic unit (esu). It is defined as follows: Two point charges, each of 1 esu and separated by \(1 \mathrm{~cm}\), exert a force of exactly 1 dyne on each other: 1 dyne \(=1 \mathrm{~g} \mathrm{~cm} / \mathrm{s}^{2}=1 \cdot 10^{-5} \mathrm{~N}\). a) Determine the relationship between the esu and the coulomb. b) Determine the relationship between the esu and the elementarv charoe

Short answer

Answer: (a) 1 esu is approximately equal to 3.34 x 10^-10 C, and (b) 1 esu is approximately equal to 4.8 x 10^-10 elementary charges.

Step-by-step solution

(Step 1: Recalling the definition of esu)

An electrostatic unit (esu) is defined as a unit of charge such that two point charges, each of 1 esu and separated by 1 cm, exert a force of exactly 1 dyne on each other. Additionally, 1 dyne is equal to \(1\times 10^{-5}\) N.

(Step 2: Using Coulomb's law)

Coulomb's law states that the electrostatic force (F) between two charged particles is given by: $$ F = k {\frac{q_1 \cdot q_2}{r^2}} $$ Where \(k\) is Coulomb's constant (\(8.99 \times 10^{9} \dfrac{Nm^2}{C^2}\)), \(q_1\) and \(q_2\) are the respective charges, and \(r\) is the distance between the charges. For our case, the given force (\(F_{esu}\)) is 1 dyne (\(1\times 10^{-5}\) N), \(q_1=q_2=1 \text{ esu}\), and \(r=1 \text{ cm}=0.01 \text{ m}\).

(Step 3: Solve for the relationship between esu and coulomb)

Now we plug the values into Coulomb's law formula to find the relationship between esu and coulomb. $$1\times 10^{-5} \mathrm{N} = k \frac{(1 \text{ esu})^2}{(0.01)^2} $$ Solving for esu: $$ 1 \text{ esu} = \sqrt{\frac{1\times 10^{-5} \mathrm{N} \cdot (0.01)^2}{k}} $$ Plug in the value of \(k\) to get the conversion factor between esu and coulomb: $$ 1 \text{ esu} = \sqrt{\frac{1\times 10^{-5} \mathrm{N} \cdot (0.01)^2}{8.99 \times 10^{9} \dfrac{Nm^2}{C^2}}} $$ Calculating this, we get: $$ 1 \text{ esu} \approx 3.34\times 10^{-10} \text{ C}$$

(Step 4: Determine the relationship between esu and elementary charge)

We know that the elementary charge (\(e\)) is equal to \(1.6 \times 10^{-19} \text{ C}\). To find the relationship between esu and the elementary charge, we will divide the elementary charge by the conversion factor found in Step 3: $$ \frac{e}{1 \text{ esu}} = \frac{1.6 \times 10^{-19} \text{ C}}{3.34\times 10^{-10} \text{ C}} $$ Calculating this, we get: $$ \frac{e}{1 \text{ esu}} \approx 4.8\times 10^{-10} $$ So, the relationship between the esu and the elementary charge is: $$ 1 \text{ esu} \approx 4.8\times 10^{-10} \text{ elementary charges} $$

Key concepts

Coulomb's Law

Coulomb's law is a fundamental principle in physics that describes the electrostatic interaction between charged particles. According to this law, the force (\(F\)) between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. This can be expressed mathematically as:

• \[ F = k \frac{q_1 \cdot q_2}{r^2} \]

Where:

• \(F\) is the electrostatic force,
• \(k\) is Coulomb's constant (approximately \(8.99 \times 10^9 \dfrac{Nm^2}{C^2}\)),
• \(q_1\) and \(q_2\) are the point charges,
• \(r\) is the distance between the charges.

In essence, Coulomb's law allows us to calculate the magnitude of the force between charges at a given distance. The interaction is attractive if the charges are opposite and repulsive if they are the same. Understanding this law helps us explore the nature of electrical forces in various contexts.

Elementary Charge

The elementary charge is a fundamental constant in physics that represents the smallest unit of electric charge that is indivisible. It is symbolized by \(e\) and its value is approximately \(1.6 \times 10^{-19} \text{ C}\).
This charge is carried by particles like protons and electrons, with a proton having a positive elementary charge and an electron having a negative one of equal magnitude.
Understanding the elementary charge is crucial in quantifying electrical interactions and explaining atomic structure. It serves as the basic unit to measure charge in the International System of Units (SI), making calculations related to charge simple and consistent. Its significance extends to understanding phenomena like electric current, where a flow of elementary charges constitutes current.

Unit Conversion

Unit conversion in physics allows us to express quantities in different systems, which is necessary for applying theoretical principles to practical scenarios. In the context of electric charge, converting from electrostatic units (esu) to coulombs (C) is essential for reconciling different measurement systems.
As demonstrated in the original exercise, we derived the relationship:

• \(1 \text{ esu} \approx 3.34 \times 10^{-10} \text{ C}\)

This conversion is crucial because the coulomb is the standard unit of charge in the SI system. Moreover, using this conversion, we find the elementary charge in terms of esu, indicating how many times it fits into a single elementary charge:

• \(1 \text{ esu} \approx 4.8 \times 10^{-10} \text{ elementary charges}\)

Understanding these conversions ensures consistency across scientific analyses and experiments, making it possible to communicate and collaborate globally using a unified standard.