Two identical positive point charges, each of magnitude q = +2.5 \times 10^{-6}\text{ C}, … — JAMB UTME

Two identical positive point charges, each of magnitude $q = +2.5 \times 10^{-6}\text{ C}$ , are fixed in a vacuum at a distance of $0.60\text{ m}$ apart. A third point charge $q_0 = +1.0 \times 10^{-6}\text{ C}$ is placed on the perpendicular bisector of the line joining the two fixed charges, at a distance of $0.40\text{ m}$ from their midpoint. Taking Coulomb's constant $k = 9.0 \times 10^9\text{ N m}^2\text{ C}^{-2}$ , what is the magnitude of the net electrostatic force acting on the third charge, in newtons?

Answer: 0.144 N

Answer

The magnitude of the net electrostatic force acting on the third charge is

0.144\text{ N}

Each fixed charge exerts an equal repulsive electrostatic force of

0.09\text{ N}

on the third charge. Due to the symmetrical arrangement, the force components perpendicular to the bisector cancel each other out, while the parallel components add together, giving a net force of

2 \times 0.09 \times 0.8 = 0.144\text{ N}

Step-by-Step Solution

Determine the distance from each fixed charge to the third charge

r = 0.50\text{ m}

The charges form a right-angled triangle with base

0.30\text{ m}

(half of

0.60\text{ m}

) and height

0.40\text{ m}

, yielding a hypotenuse of

\sqrt{0.30^2 + 0.40^2} = 0.50\text{ m}

Calculate the magnitude of the individual repulsive force from one charge

F = 0.09\text{ N}

Applying Coulomb's law:

F = \frac{k q q_0}{r^2} = \frac{9.0 \times 10^9 \times 2.5 \times 10^{-6} \times 1.0 \times 10^{-6}}{0.25} = 0.09\text{ N}

Determine directional component of forces along the perpendicular bisector

\cos\theta = 0.8

The directional cosine along the axis of symmetry is the ratio of the adjacent side (

0.40\text{ m}

) to the hypotenuse (

0.50\text{ m}

Compute the net electrostatic force using vector addition

F_{\text{net}} = 0.144\text{ N}

Horizontal components cancel by symmetry, so

F_{\text{net}} = 2 F \cos\theta = 2 \times 0.09 \times 0.8 = 0.144\text{ N}

Key Concept

Vector superposition of Coulombic forces along an axis of symmetry

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