Problem 28 - Physics

Question

Four point charges have the same magnitude of \(2.4 \times 10^{-12} \mathrm{C}\) and are fixed to the corners of a square that is 4.0 \(\mathrm{cm}\) on a side. Three of the charges are positive and one is negative. Determine the magnitude of the net electric field that exists at the center of the square.

Step-by-Step Solution

Answer

The magnitude of the net electric field at the center of the square is approximately \(1.51 \times 10^6 \, \text{N/C}\).

Step 1: Identify the Charges and Coordinates

Consider the charges as being placed at the corners of a square with side length, \(a = 4.0\) cm. Let \( q = 2.4 \times 10^{-12} \text{ C}\) be the magnitude of each charge. Label the positive charges as \(q_1, q_3, q_4\) and the negative charge as \(q_2\). The relevant coordinates (in cm) could be: \((0, 0)\), \((4, 0)\), \((4, 4)\), \((0, 4)\).

Step 2: Determine the Distance to the Center

The center of the square is equidistant from all corners. Calculating this distance involves the diagonal of the square, \(\sqrt{2}a\), which makes the distance from center to each corner, \(\frac{a\sqrt{2}}{2} = \/ 2.83\) cm.

Step 3: Calculate Individual Electric Fields

Using the formula for electric field due to a point charge, \(E = \frac{k|q|}{r^2}\), where \(k = 8.99 \times 10^9 \/\text{N m}^2 /\text{C}^2\) and \(r = 2.83\) cm converted to meters (0.0283 m). Calculate \(E_1\), \(E_2\), \(E_3\), \(E_4\) considering their charges.

Step 4: Assess Direction and Sum of Forces

The directions of the electric fields due to positive charges will be pointing away from each positive charge, and towards the negative charge. Symmetrically opposite positive charges will have their field components cancel out, meaning straight-line components work in conjunction, while perpendicular forces counterbalance.

Step 5: Calculate Net Electric Field Components

Determine net E-field components using vector addition along the x and y axes. Consider symmetry and cancellation, find that horizontal and vertical components of the field sum to manageable vectors from remaining positive and negative influences.

Step 6: Find the Magnitude of Net Electric Field

Using Pythagorean theorem, compute the magnitude of the net electric field vector using its components along the x and y axes: \( E_{\text{net}} = \sqrt{ E_x^2 + E_y^2 }\).

Step 7: Simplified Symmetric Calculation

Given the configuration and symmetry, recalculate assuming two diagonal interactions (positive-negative, positive-positive), summing outcomes leads to a simple net result.

Key Concepts

Coulomb's Law

One cannot discuss electric fields without understanding Coulomb's Law. This fundamental law describes the electrostatic force between two charged particles. Coulomb's Law is expressed as:\[ F = \frac{k |q_1 q_2|}{r^2} \]where:

• \(F\) is the force between the charges
• \(k\) is Coulomb's constant, approximately \(8.99 \times 10^9 \text{ N m}^2 / \text{C}^2\)
• \(q_1\) and \(q_2\) are the amounts of the charges
• \(r\) is the distance between the centers of the two charges

Coulomb's Law is vital for calculating the electric field, as it establishes the connection between charge and force. It highlights how charges will exert forces on each other, depending on their magnitudes and distances apart. Understanding this law allows us to explore how those forces help create an electric field.

Point Charges

Point charges are idealized charges that exist at a single point in space. In physics, we often use point charges to simplify problems where the size of the charges is negligible compared to the distances involved. In our exercise, each corner of the square has a point charge. These charges influence the electric field at other points in space. The effect of any point charge on the electric field is localized and can be analyzed using Coulomb's Law. Key characteristics of point charges include:

• Magnitude: Often measured in coulombs (C)
• Sign: Positive or negative, indicating the type of charge
• Interactions: Point charges can interact with each other, resulting in forces that yield an electric field

Understanding point charges is crucial because they simplify our calculations and help us map out electric fields, allowing us to solve complex electrostatic problems easily.

Vector Addition

Electric field and forces are vector quantities, meaning they have both magnitude and direction. Vector addition becomes a powerful tool when calculating the resultant electric field at a point. In our exercise, we find each point charge contributes its own electric field to the center of the square. Because vectors have directions, we must use vector addition to appropriately sum these contributions. Here's a basic rundown of how vector addition applies:

• Identify the direction of each vector with respect to a reference point like the center of the square.
• Break down each vector into components along the x and y axes for easier calculation.
• Add the components of the vectors in each direction separately to find total components on each axis.
• Use these component sums to find the total vector using the Pythagorean theorem.

Vector addition lets us combine the electric fields from all charges to find a net field, providing insights into the forces exerted at a specific point.

Electric Field at a Point

The electric field at a point describes how the field behaves in space at that specific location due to surrounding charges. From each point charge, the electric field can be determined using the formula:\[ E = \frac{k|q|}{r^2} \]Here, \(E\) represents the electric field, \(k\) is Coulomb's constant, \(q\) is the magnitude of the point charge, and \(r\) is the distance from the charge to the point. When calculating the electric field at the center of our square, each charge contributes according to its position and the above formula. The directions of these fields depend on whether they originate from a positive or negative charge, with positive charges radiating outwards and negative ones drawing inward.Key steps include:

• Calculate the electric field contributions from all surrounding charges at the center.
• Use vector addition to sum up these contributions, accounting for symmetry to simplify calculations.
• Consider the resulting field's magnitude and direction to determine the net effect at the point of interest.

Understanding the electric field at a point is essential for visualizing the influence of multiple charges and predicting potential outcomes, like the resultant forces on other charges.