Problem 54 - University Physics with Modern Physics

Question

Identical charges \(q = +\)5.00 \(\mu\)C are placed at opposite corners of a square that has sides of length 8.00 cm. Point \(A\) is at one of the empty corners, and point \(B\) is at the center of the square. A charge \(q_0 = -\)3.00 \(\mu\)C is placed at point \(A\) and moves along the diagonal of the square to point \(B\). (a) What is the magnitude of the net electric force on \(q_0\) when it is at point \(A\)? Sketch the placement of the charges and the direction of the net force. (b) What is the magnitude of the net electric force on \(q_0\) when it is at point \(B\)? (c) How much work does the electric force do on \(q_0\) during its motion from \(A\) to \(B\)? Is this work positive or negative? When it goes from \(A\) to \(B\), does \(q_0\) move to higher potential or to lower potential?

Step-by-Step Solution

Answer

The magnitude of force at point \( A \) is approximately \( 29.78 \, \text{N} \) directed diagonally across the square; at point \( B \), the net force is \( 0 \, \text{N} \); no work is done moving from \( A \) to \( B \). Potential remains unchanged.

Step 1: Understanding Charge Placement

Two identical charges, each with charge \( q = +5.00 \, \mu\text{C} \), are placed at opposite corners of a square with side length \( 8.00 \, \text{cm} \). A negative charge \( q_0 = -3.00 \, \mu\text{C} \) starts at one of the empty corners, point \( A \), and moves to the center of the square, point \( B \).

Step 2: Calculating Net Force at Point A

To find the net force at point \( A \), calculate the force due to each positive charge separately and use vector addition. The distance from each positive charge to \( A \) is \( 8.00 \, \text{cm} \) or \( 0.08 \, \text{m} \).Use Coulomb's law: \[ F = \frac{k |q_0| |q|}{r^2} \]where \( k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \), \( r = 0.08 \, \text{m} \), and \( q = 5.00 \, \mu\text{C} \). Calculate the magnitude of the force: \[ F = \frac{(8.99 \times 10^9) (3.00 \times 10^{-6}) (5.00 \times 10^{-6})}{(0.08)^2} \approx 21.06 \, \text{N} \] The forces from the two charges form a resultant vector diagonally across the square, found using the Pythagorean theorem as \( F_{net} = \sqrt{2} \cdot 21.06 \, \text{N} \).

Step 3: Analyzing Force Direction at A

Since both forces are away from the positive charges and point at an angle of \( 45^\circ \) with the sides of the square, the net force direction is along the diagonal from the charge at \( A \) through the center of the square, towards the diagonal of the opposite charges.

Step 4: Calculating Net Force at Point B

At point \( B \), which is equidistant from the corners of the square, the forces exerted by both positive charges are equal in magnitude and opposite in direction; thus, they cancel out completely, resulting in a net force of \( 0 \, \text{N} \).

Step 5: Calculating Work Done from A to B

Work done by the electric force is given by \( W = q_0(V_B - V_A) \), where \( V \) is the electric potential. Since the charges are oppositely equal charges on a square, \( V_B = V_A \), so the work done, \( W = 0 \).

Step 6: Determining Change in Potential

Since no work is done moving \( q_0 \) from \( A \) to \( B \), there is no change in electric potential energy (\( V_B = V_A \)), and hence, no change to higher or lower potential.

Key Concepts

Coulomb's Law

Coulomb's law is a fundamental principle used to calculate the electric force between two point charges. The force exerted by each charge on the other is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
This is expressed mathematically as:

• \( F = \frac{k |q_1| |q_2|}{r^2} \)

Here, \( F \) represents the magnitude of the force, \( k \) 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 charges, and \( r \) is the distance between them.

In our exercise, this law helps us calculate the forces exerted by the positive charges at opposite corners of the square on the negative charge that moves along the diagonal. Knowing this allows us to determine the net force on the charge at different points.

Electric Potential

Electric potential is a measure of the potential energy per unit charge at a point in an electric field. It is often described as the work done by an external force to move a charge from infinity to that point, without acceleration.
The potential difference between two points \( V_A \) and \( V_B \) can be found using the formula:

• \( V = \frac{k q}{r} \)

To move a charge between two points in an electric field, the potential difference will tell us how much work is required or released.

In this problem, when the charge \( q_0 \) moves from point \( A \) to point \( B \), the electric potential at both points is the same due to symmetry. Thus, the potential difference is zero and, consequently, no net work is done.

Work Done by Electric Force

The work done by an electric force is related to the electric potential and the charge being moved. Work is given by the equation:

• \( W = q_0 (V_B - V_A) \)

Where \( W \) is the work done, \( q_0 \) is the charge being moved, and \( V_A \) and \( V_B \) are the electric potentials at points A and B, respectively.

In our example, since the potentials at point A and B are identical, the change in potential is zero. Thus, the work done in moving the charge from A to B is also zero. Understanding this concept highlights how work is only done when there is a change in potential energy.

Charge Interaction

Charge interaction involves the forces that charges exert on one another. Like charges repel, while opposite charges attract, following Newton's third law of motion—every action has an equal and opposite reaction.
When multiple charges are involved, like in this exercise, the interactions can be complex. The net force on a charge results from the vector sum of all individual forces.

• The forces from the positive charges on the negative charge \( q_0 \) at point A lead to a net force along the diagonal.
• When \( q_0 \) is in the center (point B), these forces cancel out due to symmetry, resulting in zero net force.

These interactions nicely illustrate the principles of superposition and symmetry in electrostatics, emphasizing how charge positions influence the resultant forces.