Problem 23 - University Physics with Modern Physics

Question

A point charge, \(q=4.00 \cdot 10^{-9} \mathrm{C},\) is placed on the \(x\) -axis at the origin. What is the electric field produced at \(x=25.0 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

Answer: The electric field produced by the point charge at a distance of 25.0 cm is 361.9 N/C.

Step 1: Identify given variables

We are given the following information: - Point charge, \(q = 4.00 \cdot 10^{-9} \mathrm{C}\) - Distance from the point charge, \(x = 25.0 \mathrm{cm}\), which we need to convert to meters: \(x = 0.25 \mathrm{m}\) Our goal is to find the electric field, \(E\), at this distance.

Step 2: Use Coulomb's Law to find the electric field

Coulomb's Law for the electric field produced by a point charge is given by the formula: \(E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}\) Where: - \(E\) is the electric field - \(\epsilon_0\) is the vacuum permittivity (electric constant), with a value of \(8.85 \cdot 10^{-12} \mathrm{C^2/N} \cdot \mathrm{m^2}\), - \(q\) is the charge of the point charge - \(r\) is the distance from the point charge to the point where we want to find the electric field. In this exercise, we will use \(x\) instead of \(r\) for distance.

Step 3: Substitute the given values and calculate the electric field

We will now plug in the given values to find the electric field at the desired \(x\): \(E = \frac{1}{4\pi(8.85 \cdot 10^{-12}\,\mathrm{C^2/N} \cdot \mathrm{m^2})} \cdot \frac{4.00 \cdot 10^{-9}\,\mathrm{C}}{(0.25\, \mathrm{m})^2}\) Calculate the electric field: \(E = 361.9\,\mathrm{N/C}\)

Step 4: Write the final answer

The electric field produced by the point charge at \(x = 25.0\,\mathrm{cm}\) is \(361.9\,\mathrm{N/C}\).

Key Concepts

Coulomb's Law

Coulomb's Law describes how electric forces interact between charged objects. This important law is named after Charles-Augustin de Coulomb, a French physicist, who formulated it in 1785. It explains the force between two point charges. The electric force

• is directly proportional to the product of the magnitudes of the charges
• is inversely proportional to the square of the distance between them

In mathematical terms, for calculating the electric field from a point charge, Coulomb's Law can be expressed as:\[E = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{r^2}.\]Here, \(E\) is the electric field caused by a point charge \(q\), and \(r\) is the distance between the charge and the point of interest. Notice how the relationship is easier to manage due to the constants and square of distance, indicating sensitivity to how close or far the charges are placed. This approach gives a clear method to predict electric field strength by simply knowing the amount of charge and distance.

Point Charge

A point charge is an idealized model in electromagnetism, representing a charge located at a single point in space. This simplification serves to simplify complex real-world charges, allowing easier calculations of electric fields and forces.

• It can be a single electron or a proton that has a very small size compared to the distances in the problem.
• A point charge provides a pure charge source with no distribution area or volume.
• This idealization helps in applying fundamental equations such as Coulomb's Law without considering the geometric complexities of real objects.

Working with point charges allows students and professionals to effectively predict and understand the behavior of electrical fields in numerous scenarios. It assumes that the charge's effective field is disrupted or changed very little due to the physical size of the charge. This concept often acts as a building block for more complex electric field concepts.

Vacuum Permittivity

Vacuum permittivity, often denoted as \(\epsilon_0\), is a fundamental physical constant crucial in electromagnetism. Also called the electric constant, \(\epsilon_0\) characterizes the ability of a vacuum to permit electric field lines.

• Its value is approximately \(8.85 \times 10^{-12} \text{ C}^{2}/\text{N} \cdot \text{m}^2\).
• This constant appears frequently in equations governing electric phenomena, including Coulomb's Law.
• It provides a scale factor for measuring the electric effects in spaces devoid of matter, essentially setting the "baseline" around which electromagnetic laws are applied and measured in vacuum conditions.

Understand that vacuum permittivity links the force between charges and their separation distance, helping to quantify the field strength in theoretical and practical calculations. This constant encapsulates the inherent property of space itself in relation to electric fields.