A friend is working on a biology experiment and needs to create an electric field of magnitude \(430 \mathrm{N} / \mathrm{C}\) at \(10 \mathrm{cm}\) from the central portion of a large nonconducting square plate 4.5 m on each side. She needs to know how much charge to put on the plate. What do you tell her?

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charges: positive and negative. Like charges repel each other, while opposite charges attract. The unit of electric charge is the coulomb (C), which is equivalent to the charge carried by approximately 6.242 x 10¹⁸ protons.

The concept of electric charge is crucial in many fields of science, especially in physics and electronics. In our textbook exercise, the problem requires calculating the amount of charge needed to create a specific electric field. It’s important to understand that the charge is distributed over the surface of the plate, which affects the resulting electric field. The charge can be calculated using the electric field formula and the permittivity of free space, a constant value that provides a measure of the ability of the vacuum to permit electric field lines.

Permittivity of free space, denoted by the symbol \(\varepsilon_0\), is a physical constant that describes how an electric field affects and is affected by a vacuum. Also known as the electric constant, its value is approximately 8.85 x 10^-12 C²/N \(\cdot\) m².

This constant is essential when calculating the force between two point charges in a vacuum, as well as the electric field created by a charge in a vacuum. In our exercise, we use this constant in the electric field formula to solve for the charge required on a nonconducting square plate to achieve a certain electric field strength. Understanding the role of \(\varepsilon_0\) helps to grasp how charges interact in free space without any material medium.

The electric field formula is an equation that relates the electric field strength to the charge that produces it and the permittivity of free space. For a plane like the large square plate in our problem, it’s described as \(E = \frac{q}{2 \varepsilon_0}\), where \(E\) is the electric field strength, \(q\) is the charge, and \(\varepsilon_0\) is the permittivity of free space.

In the solution, this formula is re-arranged to calculate the charge, with \(q = E \times 2 \varepsilon_0\). When substituting the known values into this formula, you can solve for the charge required to achieve a particular electric field at a certain distance from the plate. This rearrangement is a powerful technique in physics, allowing you to isolate and calculate unknown variables given certain parameters.

In application, the formula can become more complex depending on the geometry of the charged object and the point in space where the electric field is measured. Nevertheless, the basic principle remains the same and is pivotal for a wide range of applications, including electrical engineering and fundamental physics research.