Three identical charged balls each of charge $2\mathrm{C}$ are suspended from a common point P by silk threads of $2 \mathrm{m}$ each (as shown in figure). They form an equilateral triangle of side $1 \mathrm{m}$.

The ratio of net force on a charged ball to the force between any two charged balls will be :

 

 

 

 

We are tasked with finding the ratio of the net force on a charged ball to the force between any two charged balls. Let's analyze step-by-step.

Step 1: Force Between Any Two Charges

The three charged balls form an equilateral triangle, and the charge on each ball is 2 C. Using Coulomb's law, the electrostatic force between any two charges is:

F = k * (q₁ * q₂) / r²

Substituting the given values:

• k: Coulomb's constant
• q₁ = q₂ = 2 C
• r = 1 m (distance between any two charges)

F = k * (2 * 2) / (1)² = 4k

Step 2: Net Force on One Charge

Each charged ball experiences forces due to the other two balls. Since the configuration is symmetric (equilateral triangle), the angle between the forces is 120°. Resolving these forces:

• Each force magnitude is F = 4k.
• Resolve the two forces into components. The net force lies along the bisector of the angle.

The horizontal components cancel out, while the vertical components add up:

F_net = 2 * F * cos(30°)

Substituting the value of cos(30° = √3 / 2:

F_net = 2 * 4k * (√3 / 2) = 4k√3

Step 3: Ratio of Net Force to Force Between Two Charges

The ratio is given by:

Ratio = F_net / F

Substituting the values:

Ratio = (4k√3) / (4k) = √3

Final Answer

The ratio of the net force on one charged ball to the force between any two charged balls is √3.

Remark

While calculating the forces, it is important to note that due to symmetry, the net horizontal force is zero. However, the net vertical component determines the net force magnitude. If options in the question do not match, this may indicate an issue with the options provided.