Problem 69 - College Physics

Question

\(\bullet$$\bullet\) Point charges of 3.00 \(\mathrm{nC}\) are situated at each of three cor- ners of a square whose side is 0.200 \(\mathrm{m} .\) What are the magni- tude and direction of the resultant force on a point charge of \(-1.00 \mu \mathrm{C}\) if it is placed (a) at the center of the square, (b) at the vacant corner of the square?

Step-by-Step Solution

Answer

(a) 0 N at the center; (b) 1.292 mN to the square's center when at the corner.

Step 1: Understanding the Problem

We have four corners of a square, three of which have point charges of 3.00 nC. We need to find the force on a test charge of -1.00 µC placed either at the center or at the vacant corner of the square.

Step 2: Force at the Center of the Square

When the -1.00 µC charge is at the center, symmetry shows that the resultant force from the three 3.00 nC charges cancels out. This is because the attractions from the three corners act in directions that add to zero due to symmetry.

Step 3: Force on the Charge at the Corner

Now, consider the -1.00 µC charge placed at the vacant corner. Calculate the force due to each 3.00 nC charge on this -1.00 µC charge. Use Coulomb's Law for each pair: \[ F = k_e \frac{|q_1 q_2|}{r^2} \]where \( k_e \) is Coulomb's constant \( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \), \( q_1 = 3.00 \times 10^{-9} \) C, \( q_2 = -1.00 \times 10^{-6} \) C, and \( r = 0.200 \) m.

Step 4: Calculating Individual Forces

For each pair of charges at the adjacent corners, calculate the force. Since there are two adjacent 3.00 nC charges, compute:\[ F_{adj} = k_e \frac{3.00 \times 10^{-9} \cdot 1.00 \times 10^{-6}}{(0.200)^2} \approx 6.7425 \times 10^{-4} \, \text{N} \]For the opposite corner, where the diagonal is the distance, use \( r = \sqrt{2} \times 0.200 \):\[ F_{opp} = k_e \frac{3.00 \times 10^{-9} \cdot 1.00 \times 10^{-6}}{(0.200\sqrt{2})^2} \approx 3.3712 \times 10^{-4} \, \text{N} \].

Step 5: Determining Resultant Force

The forces at the adjacent corners have the same magnitude and are at right angles, so use vector addition to find their resultant:\[ F_{adj, resultant} = \sqrt{(6.7425 \times 10^{-4})^2 + (6.7425 \times 10^{-4})^2} \approx 9.54 \times 10^{-4} \, \text{N} \] This resultant from adjacent corners combines linearly with the force from the opposite corner (since it's along the diagonal), to get the total force.

Step 6: Calculating Total Force Magnitude and Direction

Add the resultant of adjacent forces to the opposite force for magnitude:\[ F_{total} = 9.54 \times 10^{-4} + 3.37 \times 10^{-4} = 1.292 \times 10^{-3} \, \text{N} \]The direction is along the diagonal from the vacant corner to the center of the square.

Key Concepts

Coulomb's Law

Coulomb's Law is fundamental in electrostatics, describing how two charges interact. It quantifies the force between electric charges using the formula: \[ F = k_e \frac{|q_1 q_2|}{r^2} \ \] where

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

This law implies that the force between two point charges is directly proportional to the product of their charge magnitudes and inversely proportional to the square of the distance between them.
The direction of the force is along the line joining the charges. Like charges repel, while unlike charges attract.

Vector Addition

In physics, particularly electrostatics, vector addition helps in determining the net force when multiple forces are acting on a charge. Types of vector addition commonly used include graphical and analytical methods.
For analytical vector addition, each vector is broken into its horizontal and vertical components based on their direction and magnitude.
Consider two forces, \( F_1 \) and \( F_2 \), represented as vectors:

• Calculate the components along the x-axis and y-axis.
• Add the respective components across all vectors.
• Find the resultant vector \( \vec{F}_{resultant} \) using the Pythagorean theorem:

\[ F_{resultant} = \sqrt{(F_{x})^2 + (F_{y})^2} \ \] This process assists in understanding the overall effect of multiple adjacent forces acting upon a point charge.

Symmetry in Electrostatics

Symmetry plays a significant role in electrostatic problems. When charges are symmetrically placed, it simplifies the calculations significantly.
In the exercise provided, placing the test charge at the center of the square results in the forces from each corner charge canceling due to symmetry.

• Each charge exerts equal force magnitude on the test charge, but in mutually opposite directions.
• This results in net forces that cancel each other out, leading to a zero resultant force at the center of symmetry.

When symmetry is present, it can dramatically reduce computational complexity, allowing for a straightforward calculation of resultant forces.

Resultant Force Calculations

Resultant force calculations involve determining the net effect of multiple interacting forces on a point charge. In the context of our exercise, this process involves:

• Calculating the force due to each individual point charge acting on the test charge using Coulomb's Law.
• Using vector addition to combine these forces into a single resultant force.
• The forces at adjacent points are summed vectorially using the Pythagorean theorem.

Given their perpendicular relationship, their magnitudes form the sides of a right triangle, leading to a straightforward vector sum of:\[F_{adj, resultant} = \sqrt{(F_{adj_x})^2 + (F_{adj_y})^2}\]
Finally, adding this result with the diagonal force completes the overall force calculation. Understanding this process is essential for resolving force interactions in electrostatic fields efficiently.