Question

Four charges \(Q_{1}, Q_{2}, Q_{3}\) and \(Q_{4}\) of same magnitude are fixed along the \(x\) axis at \(x=-2 a\), \(-a,+a\) and \(+2 a\), respectively. A positive charge \(q\) is placed on the positive \(y\) axis at a distance \(b>0\). Four options of the signs of these charges are given in List \(\mathrm{I}\). The direction of the forces on the charge \(q\) is given in List II. Match List I with List II and select the correct answer using the code given below the lists. List | P. \(Q_{1}, Q_{2}, Q_{3}, Q_{4}\) all positive Q. \(Q_{1}, Q_{2}\) positive; \(Q_{3}, Q_{4}\) negative R. \(Q_{1}, Q_{4}\) positive; \(Q_{2}, Q_{3}\) negative S. \(Q_{1}, Q_{3}\) positive; \(Q_{2}, Q_{4}\) negative List II \(1 .+x\) 2\. \(-x\) \(3 .+y\) 4\. \(-y\) (A) \(\mathrm{P}-3, \mathrm{Q}-1, \mathrm{R}-4, \mathrm{~S}-2\) (B) \(\mathrm{P}-4, \mathrm{Q}-2, \mathrm{R}-3, \mathrm{~S}-1\) (C) P-3, Q-1, R-2, S-4 (D) \(P-4, Q-2, R-1, S-3\)

Short answer

The correct match is (A) P-3, Q-1, R-4, S-2.

Step-by-step solution

Analyzing the Charge Configurations

Determine the direction of the electric force exerted by each charge on the positive charge q due to Coulomb's Law. Charges with the same sign repel and charges with opposite signs attract. Given that q is positive, a positive charge will exert a force on q radially away from itself (repulsion), and a negative charge will exert a force toward itself (attraction).

Summing Forces for Configuration P

Since all charges are positive, they will all repel the positive charge q. The forces from Q1 and Q4 will have horizontal components canceling each other out, and the same occurs for Q2 and Q3. The net force will be along +y direction, away from the x-axis.

Summing Forces for Configuration Q

Q1 and Q2 will repel q, while Q3 and Q4 will attract it. The horizontal components of force due to Q1 and Q4 as well as Q2 and Q3 will cancel out. The vertical components of force from Q1 and Q2 (repelling) will be greater than the vertical components of force from Q3 and Q4 (attracting) because they are closer to q. The net force is along the +x direction.

Summing Forces for Configuration R

Q1 will repel q, and Q4 will attract q, which results in a net force to the right along the +x axis. Q2 will repel and Q3 will attract with a smaller force because Q3 is farther away, resulting in a net force to the left. The horizontal components of forces from Q1 to Q4 cancel each other, and the net force is along the -y direction.

Summing Forces for Configuration S

Q1 will repel q while Q3 will attract it, resulting in a net force downward along the -y axis since Q3 is closer and the y-components of their forces will not cancel out. Q2 will repel and Q4 will attract with horizontal components canceling each other out. The net force is along the -x direction.

Matching Configurations to Directions

Match each configuration P, Q, R, S with the corresponding direction from List II. P corresponds with 3 (+y), Q corresponds with 1 (+x), R corresponds with 4 (-y), and S corresponds with 2 (-x).

Key concepts

Electric Charge Configurations

Electric charge configurations play a pivotal role in understanding the behavior of charges and the resulting electric forces. In a scenario where multiple charges are arranged in a certain way, the overall influence on a test charge depends on not just the magnitude but also the signs and relative positions of the other charges. For the four charges fixed along the x-axis and the positive charge q placed on the y-axis, each configuration has a unique outcome on the forces acting on charge q.

For students tackling such problems, it's imperative to visualize the arrangement and consider symmetry in the configurations. Charges with the same sign will induce forces in the same direction (repulsion if both are positive) if placed symmetrically around charge q. These forces can be broken down into horizontal and vertical components. In a well-ordered setup, like the one described, some of these components will naturally cancel each other out due to symmetry, simplifying the task of finding the net force on charge q.

Forces Between Charges

Coulomb's Law quantifies the forces between two point charges, stating that the magnitude of the electric force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. But it's not just the magnitude; the direction is equally critical. As per the law, unlike charges attract, and like charges repel. When evaluating forces between charges, like in our textbook exercise, calculate the resultant force vector by considering both magnitude and direction of the individual forces.

For instance, take configuration Q from the exercise, where charges Q1 and Q2 on the x-axis are positive and closer to q than the negative charges Q3 and Q4. Here, we need to remember the superposition principle—the net force is the vector sum of individual forces. This means we calculate and then combine the forces exerted by each charge on q separately, keeping in mind that positive charges repel and negative charges attract, leading to forces in opposite directions.

Electric Force Direction

Determining the electric force direction is essential to solving any Coulomb's Law problem. It is based on the fundamental nature of interactions between charges: like charges repel, and opposite charges attract. The direction of the electric force will always be along the line connecting the centers of the two charges involved in the interaction.

When considering more complex scenarios with multiple charges, the direction of the net electric force acting on a particular charge is the vector sum of the forces due to all other charges. In terms of the exercise, we analyze the net force direction by considering the signs of the charges and their respective directions of interaction with the test charge q. The careful consideration of each force's direction is the key to matching the charge configurations with the correct resultant force directions listed.