Problem 77 - A Complete Resource Book in Physics for JEE Main

Question

Two equal positive charges \(+q\) each are fixed a certain distance apart. A third equal positive charge \(+q\) is placed exactly midway between them. Then the third charge will (A) move at an angle of \(45^{\circ}\) to the line joining the two charges. (B) move at an angle of \(90^{\circ}\) to the line joining the two charges. (C) move along the line joining the two charges. (D) stay at rest.

Step-by-Step Solution

Answer

Here, we have three equal positive charges, \(+q\), with two fixed charges and a third charge placed midway between them. Since the forces exerted by the two fixed charges on the third charge are equal in magnitude but opposite in direction, they cancel each other out, and the net force acting on the third charge is zero. As the net force is zero, according to Newton's first law of motion, the third charge will stay at rest. Therefore, the answer is (D).

Step 1: Identify knowns and unknowns

We are given: -The fixed charges are equal positive charges, \(+q\). - The third charge placed between them has the same charge, \(+q\). We want to find: - The movement of the third charge.

Step 2: Coulomb's Law for electrostatic force

To analyze the movement of the third charge, we should first see the forces acting on it by Coulomb's Law, which states that: \[F = k \frac{q_1 \cdot q_2}{r^2}\] where \(F\) is the electrostatic force between the charges, \(k\) is the electrostatic constant, \(q_1\) and \(q_2\) are the charges and \(r\) is the distance between the two charges. Since the charges are like-charges (positive) and are all equal, the forces between them will be repulsive.

Step 3: Calculate electrostatic forces exerted by the fixed charges on the third charge

We know that, when both charges \(+q\) are fixed and we place the third charge exactly midway, the line connecting the two fixed charges will be perpendicular bisected. Thus, at this moment, the force exerted by one fixed charge on the third charge will have the same magnitude but opposite direction of the force exerted by the other fixed charge on the third charge. Both forces will have the same magnitude as given by Coulomb's Law because the charges are equal and at the same distances: \[F = k \frac{q^2}{r^2}\]

Step 4: Net force

Since the forces exerted by the two fixed charges on the third charge are equal in magnitude but opposite in direction, they cancel each other out, and the net force acting on the third charge is zero.

Step 5: Conclusion

As the net force acting on the third charge is zero, it will not experience any acceleration. According to Newton's first law of motion, a body at rest will remain at rest if the net force acting on it is zero. Therefore, the third charge will: (D) stay at rest. So, the answer is (D).