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 elementary charge.

Short answer

Question: Determine the relationship between the esu unit of charge and the coulomb unit, and the relationship between the esu and the elementary charge. Answer: The relationship between the esu unit of charge and the coulomb unit is 1 esu ≈ 3.336 × 10^{-10} C. The relationship between the esu and the elementary charge is 1.602 × 10^{-19} C ≈ 4.803 × 10^{-10} esu.

Step-by-step solution

Review Coulomb's Law

Coulomb's law states that the force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them, which can be represented as: \(F = k * \frac{Q_1 * Q_2}{r^2}\) where: - F is the force between the charges - k is the electrostatic constant - Q1 and Q2 are the charges - r is the distance between the charges. Take note that here, k is in units of dynes when using the electrostatic units (esu). Therefore, \(1 \, dyne = k * \frac{(1 esu)^2}{(1 cm)^2}\)

Convert the dyne to Newtons

The exercise states that 1 dyne \(=1 \cdot 10^{-5} N\). So, the equation becomes: \(1 \cdot 10^{-5} N = k * \frac{(1 esu)^2}{(1 cm)^2}\)

Convert the electrostatic constant k to SI units

In SI units, the electrostatic constant k is equal to: \(k = 8.9875 \times 10^9 N m^2 / C^2\) Now, let's express 1 esu in terms of coulombs (C). \(1 \cdot 10^{-5} N = 8.9875 \times 10^9 N m^2 / C^2 * \frac{(1 esu)^2}{(0.01 m)^2}\)

Solve for the value of 1 esu in coulombs

We will now solve for the value of 1 esu in coulombs: \((1 esu)^2 = \frac{1 \cdot 10^{-5} N * (0.01 m)^2}{8.9875 \times 10^9 N m^2 / C^2}\) \(1 esu = \sqrt{\frac{1 \cdot 10^{-5} N * (0.01 m)^2}{8.9875 \times 10^9 N m^2 / C^2}}\) \(1 esu \approx 3.336 \times 10^{-10} C\) So, the relationship between the esu and the coulomb is: \(1 esu \approx 3.336 \times 10^{-10} C\).

Determine the relationship between the esu and the elementary charge

The elementary charge (the charge of an electron or proton) is equal to: \(e = 1.602 \times 10^{-19} C\) Now, let's express the elementary charge in terms of esu: \(1.602 \times 10^{-19} C = x \, esu\) We can substitute the conversion factor we determined in step 4: \(1.602 \times 10^{-19} C = x \, (3.336 \times 10^{-10} C)\) Solve for x: \(x = \frac{1.602 \times 10^{-19} C}{3.336 \times 10^{-10} C} \approx 4.803 \times 10^{-10}\) So, the relationship between the esu and the elementary charge is: \(1.602 \times 10^{-19} C \approx 4.803 \times 10^{-10} \, esu\).

Key concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electrostatics that describes the force between two point charges. It states that the force (\f\( F \f\)) between the charges is proportional to the product of the charges (\f\( Q_1 \f\) and \f\( Q_2 \f\)) and inversely proportional to the square of the distance (\f\( r \f\)) between them. Mathematically, the law is expressed as:

\f\( F = k \times \frac{Q_1 \times Q_2}{r^2} \f\)

The constant \f\( k \f\) is known as the electrostatic force constant. When dealing with electrostatic units (esu), we see a unique setup where two charges of 1 esu each, separated by a distance of 1 cm, would exert precisely 1 dyne of force on one another. Simplifying this concept for students means framing the law in terms of real-world applications—such as explaining how the strength of the force varies with distance or how different materials can affect the constant \f\( k \f\). By understanding Coulomb's Law, students can better grasp the invisible yet significant electric forces that occur between charged particles.

Unit Conversion

Understanding unit conversion is crucial when working with different systems of measurement. In the context of electrostatics, we frequently need to convert between electrostatic units (esu) and the more commonly used SI unit, the coulomb (C).

For instance, the step-by-step solution provided calculates the value of 1 esu in coulombs (\f\( C \f\)), which helps students to navigate between these units seamlessly. It also emphasizes the importance of dimensional analysis—a technique where students ensure that the units on both sides of an equation are consistent, allowing them to confidently convert from one unit to another.

When discussing unit conversions, it is helpful to remind students that different systems exist for various historical and practical reasons and that being able to move between these systems is a key skill in scientific studies. Using real-life examples, like comparing the currency exchange rates between countries, can make this concept more relatable to students.

Elementary Charge

The concept of elementary charge is at the heart of understanding electric charge. It is the smallest unit of electric charge that is considered indivisible in classical electromagnetism and is denoted by \f\( e \f\). The charge of a single electron or a proton is approximately \f\( 1.602 \times 10^{-19} \f\) coulombs (C), and this value is fundamental to the structure of atoms.

When the exercise asks students to determine the relationship between the esu and the elementary charge, it leads them to see the link between microcosmic particles and macroscopic units of charge. The elementary charge exemplifies quantization in nature and is also integral to the concept of charge quantization, which states that the total charge of any object is an integer multiple of the elementary charge.

To make this concept more digestible, educators can use analogies such as explaining that the elementary charge is like the 'atomic unit' of charge, much as a single brick might be the fundamental unit in building a wall. With this framing, students can better appreciate the significance of the elementary charge when delving into topics like electric current and the electronic structure of atoms.