Question

Four identical point charges \((+1.61 \mathrm{nC})\) are placed at the corners of a rectangle, which measures \(3.00 \mathrm{~m}\) by \(5.00 \mathrm{~m}\). If the electric potential is taken to be zero at infinity, what is the potential at the geometric center of this rectangle?

Short answer

Question: Calculate the electric potential at the geometric center of a rectangle with four identical point charges (+1.61 nC) placed at its corners. The dimensions of the rectangle are 3.00 m (length) and 5.00 m (width). Answer: The electric potential at the geometric center of the rectangle is approximately 19.83 V.

Step-by-step solution

Understand the given data

Here, we are given four identical point charges \((+1.61 \mathrm{nC})\) placed at the corners of a rectangle. The dimensions of the rectangle are \(3.00 \mathrm{~m}\) (length) and \(5.00 \mathrm{~m}\) (width). We need to find the electric potential at the geometric center of this rectangle.

Find the distance between the charges and the center of the rectangle

In order to find the electric potential at the center of the rectangle, we first need to determine the distance between each corner point charge and the geometric center. The geometric center is located at the point where the diagonals of the rectangle intersect, which divides them into equal halves. So, the distance between the corner charges and the geometric center can be obtained using the Pythagorean theorem: \(distance = \sqrt{(\frac{length}{2})^2 + (\frac{width}{2})^2}\). Substitute the given values of length and width: \(distance = \sqrt{(\frac{3.00}{2})^2 + (\frac{5.00}{2})^2} = \sqrt{(1.5)^2+(2.5)^2} = \sqrt{2.25+6.25} = \sqrt{8.5}\).

Formula for electric potential due to a point charge

The formula for electric potential \(V\) due to a point charge \(Q\) at a distance \(r\) from the charge is given by the following equation: \(V = \frac{kQ}{r}\), where \(k\) is Coulomb's constant (\(8.99 x 10^9 \mathrm{Nm^2C^{-2}}\)).

Calculate the electric potential at the geometric center

For each charge, we'll substitute the appropriate values to calculate the electric potential, using the superposition principle. Since all charges are equal and at equal distances from the geometric center, we can find the potential due to one charge and multiply by 4 to obtain the total potential at the geometric center. Given, \(Q = +1.61 \mathrm{nC}\), converting it to coulombs, \(Q = 1.61 \times 10^{-9} \mathrm{C}\) \(r = \sqrt{8.5} \mathrm{m}\) \(k = 8.99 \times 10^9 \mathrm{Nm^2C^{-2}}\) Now, find the electric potential due to one charge using the electric potential formula: \(V_{total} = 4 \times \frac{kQ}{r} = 4 \times \frac{(8.99 \times 10^9)(1.61 \times 10^{-9})}{\sqrt{8.5}}\)

Calculate the final value of electric potential

After computation, we get: \(V_{total} \approx 4 \times \frac{(8.99 \times 10^9)(1.61 \times 10^{-9})}{\sqrt{8.5}} \approx 4 \times \frac{(14.4539 \times 10^0)}{2.9142 \mathrm{~m}} \approx 19.83 \mathrm{V}\) The electric potential at the geometric center of the rectangle is approximately \(19.83 \mathrm{V}\).

Key concepts

Point Charges

Point charges are one of the foundational concepts in electrostatics, a branch of physics which deals with the study of stationary electric charges. A point charge is an idealized model of a charged particle that has an infinitely small size, allowing it to be considered as a single point in space. Despite this simplicity, the point charge is a powerful model because it allows us to calculate electric fields and potentials created by real-life charged objects, assuming they can be approximated by points.

When dealing with problems involving point charges, such as calculating the electric potential created by charges at the corners of a geometric shape, we consider the charge to be concentrated at a single point. This makes it possible to use precise mathematical equations to determine the influence of each charge on a point of interest, such as the geometric center of a structure. The challenge often lies in calculating the distances correctly and then using these distances to find the potential due to each charge.

Electric Potential Formula

The electric potential, sometimes referred to as voltage, at a specific point in space is defined as the amount of electric potential energy per unit charge at that location. It gives us an indication of the potential energy a test charge would have if placed at that point. The formula for electric potential (V) due to a single point charge (Q) is given by:
\[V = \frac{kQ}{r}\]
where \(k\) is the Coulomb's constant (\(8.99 \times 10^9 \mathrm{Nm^2C^{-2}}\) ), \(Q\) is the charge of the point charge, and \(r\) is the distance from the point charge to the location where you want to calculate the electric potential.

In our example exercise, this formula allows us to calculate the potential at the geometric center due to four identical point charges located at the corners of a rectangle. The beauty of this formula lies in its direct proportionality to the charge and inverse proportionality to the distance, illustrating the principle that the potential will increase with greater charge and closer proximity.

Superposition Principle

The superposition principle is a critical concept when studying electric potential, as it allows us to understand and calculate the combined effect of multiple charges on electric potential at a given point in space. According to the superposition principle, the total electric potential due to a group of point charges is the algebraic sum of the potentials due to each individual charge.

This implies that the contributions to electric potential from multiple charges are additive and do not interfere with each other. In the exercise, this principle enables us to find the individual potentials at the geometric center due to each of the four point charges and then simply sum them up to find the total electric potential at the center. Since the charges and distances are all equal in this particular problem, the principle simplifies the calculation enormously. We can calculate the potential for one charge and then multiply by four to get the total potential, which streamlines our problem-solving process significantly.