three identical small balls each of mass 0.1 g are suspended at one point on silk thread having a length of 20 cm. What charges should be imparted to the balls for each thread to form an angle of alpha = 30 with the vertical.

To determine the charges that should be imparted to the three identical small balls so that each thread forms an angle of 30 degrees with the vertical, we can analyze the forces acting on the balls. This situation involves both gravitational and electrostatic forces, and we can use some principles from physics to find the solution.

Understanding the Forces at Play

Each ball experiences two main forces:

• Gravitational Force (Weight): This acts downward and is given by the formula W = mg, where m is the mass of the ball and g is the acceleration due to gravity (approximately 9.81 m/s²).
• Electrostatic Force: This acts horizontally due to the repulsion between the like charges on the balls.

Calculating the Gravitational Force

First, we need to calculate the weight of each ball:

Given that the mass of each ball is 0.1 g, we convert this to kilograms:

m = 0.1 g = 0.1 / 1000 = 0.0001 kg

Now, we can find the weight:

W = mg = 0.0001 kg × 9.81 m/s² = 0.000981 N

Analyzing the Forces at the Angle

When the balls are at an angle of 30 degrees with the vertical, we can break down the forces into components. The gravitational force can be resolved into two components:

• The vertical component: W_vertical = W cos(30°)
• The horizontal component: W_horizontal = W sin(30°)

Using the trigonometric values:

cos(30°) = √3/2 ≈ 0.866 and sin(30°) = 1/2 = 0.5

Thus, we can calculate:

W_vertical = 0.000981 N × 0.866 ≈ 0.000850 N

W_horizontal = 0.000981 N × 0.5 ≈ 0.000491 N

Equating Forces for Electrostatic Repulsion

For the balls to remain in equilibrium, the horizontal electrostatic force must equal the horizontal component of the weight:

F_electrostatic = W_horizontal

The electrostatic force between two charges is given by Coulomb's law:

F = k * |q₁ * q₂| / r²

Where:

• k is Coulomb's constant (approximately 8.99 × 10⁹ N m²/C²)
• q₁ and q₂ are the charges on the balls (since they are identical, we can denote them as q)
• r is the distance between the centers of the two balls

Finding the Distance Between the Balls

To find r, we can use the geometry of the situation. The length of the thread is 20 cm, and the angle is 30 degrees. The horizontal distance between two balls can be calculated as:

r = 2 * L * sin(30°) = 2 * 0.2 m * 0.5 = 0.2 m

Setting Up the Equation

Now we can set the electrostatic force equal to the horizontal component of the weight:

k * q² / r² = W_horizontal

Substituting the known values:

8.99 × 10⁹ N m²/C² * q² / (0.2 m)² = 0.000491 N

Solving for Charge

Rearranging the equation gives:

q² = (0.000491 N * (0.2 m)²) / (8.99 × 10⁹ N m²/C²)

q² = (0.000491 * 0.04) / (8.99 × 10⁹)

q² = 1.96 × 10⁻¹³ C²

q = √(1.96 × 10⁻¹³) ≈ 1.4 × 10⁻⁷ C

Final Thoughts

Each ball should be imparted with a charge of approximately 1.4 × 10⁻⁷ C for the threads to form an angle of 30 degrees with the vertical. This analysis combines concepts of forces, geometry, and electrostatics to arrive at a solution that reflects the balance of forces acting on the balls.