Three identical charges \(+q\) and a fourth charge \(-q\) form a square of side \(a\). (a) Find the magnitude of the electric force on a charge \(Q\) placed at the square's center. (b) Describe the direction of this force.

Understanding the electric force between charges is crucial, and this is where Coulomb's law plays a fundamental role. Coulomb's law quantifies the amount of force between two point charges. The formula is given as:
\( F = k \cdot \frac{{|q_1 \cdot q_2|}}{{r^2}} \)
where \( F \) is the magnitude of the force between the charges, \( q_1 \) and \( q_2 \) are the magnitudes of the charges involved, \( r \) is the distance between the centers of the two charges, and \( k \) is Coulomb's constant, approximately \( 8.987 \times 10^9 \) N m\textsuperscript{2}/C\textsuperscript{2}.

• The law implies that like charges repel and unlike charges attract.
• The force is inversely proportional to the square of the distance between the charges - meaning as the distance doubles, the force becomes a quarter of what it was.
• It's important to remember that Coulomb's law applies to point charges or spherically symmetric charge distributions.

In the exercise, Coulomb's law is used to determine the force exerted on the central charge \( Q \) by each of the charges at the corners of the square.

Charge interactions are at the heart of understanding how objects become electrically charged and influence each other. Two fundamental principles govern these interactions:

• Opposite charges (one positive, one negative) will attract each other.
• Similar charges (both positive or both negative) will repel each other.

In the given problem, there are three positive charges and one negative charge. The central charge \( Q \) is surrounded symmetrically by these charges. The positive charges exert forces which are in the same direction due to the square's symmetry, whereas the lone negative charge exerts a force in the opposite direction.

Understanding Symmetry in Charge Arrangement

The square arrangement of the charges creates a situation where the symmetry of the system simplifies the calculation of the net force on the central charge. Due to the identical distances and magnitudes of the three positive charges, their individual forces combine to produce a force pointing in a specific direction. The negative charge, meanwhile, generates a force in the opposite direction. Assessing these interactions and combining the forces correctly is essential to solving the problem accurately.

When dealing with multiple forces acting at a single point, it's essential to consider both the magnitude and the direction of each force, down to the concept of vectors. Vector summation allows us to combine forces that are not aligned in a straight line and find the net result.

Applying Vector Addition

Vectors have both a magnitude (how strong the force is) and a direction (where the force is heading). To sum up vectors, we need to take into account their directional influence on each other. Here's how it works in the context of the exercise:

• The total force from the three identical positive charges is calculated and directed in one way due to the symmetry of the square.
• The force from the single negative charge is calculated and directed oppositely.
• To find the net force (the resultant vector), you combine all these forces while considering their respective directions.

The step by step solution described in the exercise points out the need to subtract the force due to the negative charge from the combined forces of the positive charges because they act in opposite directions. This is a clear example of vector subtraction, which is a part of vector summation, allowing us to find the net force on the charge placed in the center of the square.