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 \(x=25.0 \mathrm{~cm} ?\)

Short answer

Answer: The electric field produced at a distance of 25.0 cm from a point charge of \(4.00 \cdot 10^{-9} \mathrm{C}\) is \(575.36 \mathrm{N} / \mathrm{m^2}\).

Step-by-step solution

Identify the known values

The known values are: - \(k = 8.99 \cdot 10^9 \mathrm{N} / \mathrm{m^2C}\), the electrostatic constant, - \(q = 4.00 \cdot 10^{-9} \mathrm{C}\), the point charge, and - \(r = 25.0 \mathrm{cm}\), which we need to convert to meters: \(0.25\mathrm{m}\).

Apply the formula for the electric field

We apply the electric field formula: \(E = \frac{k\cdot q}{r^2}\) Now, substitute the known values into the formula: \(E = \frac{(8.99 \cdot 10^9 \mathrm{N} / \mathrm{m^2C})(4.00 \cdot 10^{-9} \mathrm{C})}{(0.25\mathrm{m})^2}\)

Simplify and calculate the electric field

After substituting the values into the formula, we simplify and calculate the result: \(E = \frac{(8.99 \cdot 10^9 \mathrm{N} / \mathrm{m^2C})(4.00 \cdot 10^{-9} \mathrm{C})}{(0.0625\mathrm{m^2})}\) \(E = \frac{35.96 \cdot 10^0 \mathrm{N}}{0.0625\mathrm{m^2}}\) \(E = 575.36 \mathrm{N} / \mathrm{m^2}\) Hence, the electric field produced at \(x=25.0 \mathrm{cm}\) from the point charge is \(575.36 \mathrm{N} / \mathrm{m^2}\).

Key concepts

Point Charge

A point charge represents an idealized model of a particle with an electric charge which is considered to be concentrated at a single point in space. When calculating the electric field created by a point charge, we imagine that the entire charge exists in a space with zero volume. This simplification is widely used in physics to make the complex calculations of electric fields more manageable.

For instance, in the provided exercise, the point charge is placed at the origin of the coordinate system. Since it is a point charge, we can disregard its physical size and focus only on the magnitude of the charge and its position when calculating the electric field. This simplification is a fundamental aspect of many electrostatic problems and dynamics in electric fields.

Electrostatic Constant

The electrostatic constant, also known as Coulomb's constant (\( k \)), is a crucial value in the realm of electrostatics. It represents the force of electrical interaction between two point charges separated by one meter in a vacuum. The numerical value of the electrostatic constant is approximately \(8.99 \cdot 10^9 \mathrm{N} / \mathrm{m^2C}\).

This constant plays a vital role in calculations involving the electric field and force between charges. It is derived from Coulomb's law, which describes the electrostatic interaction, and is a measure of the strength of the electric force in a given system. In our exercise, the electrostatic constant provides the proportionality between the charge and the electric field, factoring in the distance squared between the charge and the point at which the electric field is being calculated.

Electric Field Formula

The electric field formula is an expression that defines the strength and direction of an electric field at a particular point in space. The electric field (\( E \) ) is directly proportional to the charge (\( q \) ) and inversely proportional to the square of the distance (\( r^2 \) ) from the point charge, following the equation: \[ E = \frac{k \cdot q}{r^2} \

This formula encapsulates how the presence of a point charge can influence the space around it, creating an electric field that decreases in strength with the square of the distance from the charge. By inputting the appropriate values into this formula, as seen in the exercise's step-by-step solution, one can determine the magnitude of the electric field at any given distance from a charge. The formula's simplicity allows for straightforward calculations and provides a fundamental understanding of electrostatic principles.

Coulomb's Law

Coulomb's law provides the foundational framework for understanding electrostatic interactions between two charged particles. It states that the force (\( F \) ) between two point charges is directly proportional to the product of their charges (\( q_1 \) and \( q_2 \) ) and inversely proportional to the square of the distance (\( r^2 \) ) between them, along with incorporating the electrostatic constant (\( k \) ). The law is mathematically expressed as: \[ F = \frac{k \cdot q_1 \cdot q_2}{r^2} \

Coulomb's law is vital for understanding how charged particles interact and how they create electric fields that extend into the space around them. In educational contexts, this law is often one of the first introductions to field theory for students. For the exercise that has been solved, Coulomb's law underpins the calculation method for determining the electric field, with the formula adapted to find field strength rather than force. By mastering Coulomb's law, students gain valuable insight into the behavior of electric charges and the forces they exert in space.