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Step-by-Step Solution
Step 1: Convert all quantities to SI units
• Each sphere has a mass of 10 mg = 10 × 10−3 g = 0.01 g = 10−5 kg.
• The length of each thread is L = 0.5 m.
• The distance of separation between the spheres is r = 0.20 m.
• Given gravitational acceleration g = 10 m/s2.
Step 2: Identify the geometry and forces
The spheres are suspended from the same point and repel each other horizontally. The threads make an angle θ with the vertical such that:
sin(θ) = (horizontal separation from the midpoint) / (length of thread) = (0.20/2) / 0.5 = 0.10 / 0.5 = 0.2.
Thus, θ = sin−1(0.2). We will use sin(θ) and cos(θ) in the force balance.
There are two main forces on each sphere:
Electrostatic repulsion force Fe (horizontal)
Weight mg (vertical)
Step 3: Write the force balance equations
Let T be the tension in the thread. Resolving forces:
T sin(θ) = Fe … (1)
T cos(θ) = mg … (2)
Divide (1) by (2) to eliminate T:
tan(θ) = Fe / (mg).
Therefore,
Fe = mg tan(θ).
Step 4: Calculate tan(θ)
We have sin(θ) = 0.2 ⇒ cos(θ) = √(1 − sin2(θ)) = √(1 − 0.22) = √(0.96).
Hence, cos(θ) ≈ 0.9798, and
tan(θ) = sin(θ)/cos(θ) = 0.2 / 0.9798 ≈ 0.204.
Step 5: Determine the electrostatic force Fe
mg = (10−5 kg)(10 m/s2) = 10−4 N.
Hence,
Fe = mg tan(θ) = (10−4 N)(0.204) = 2.04 × 10−5 N.
Step 6: Apply Coulomb’s Law to find the charge q
The electrostatic force between two charges q separated by distance r is given by:
Fe = (1 / 4πϵ0) (q2 / r2) = k (q2 / r2),
where k = 1/(4πϵ0) = 9 × 109 N·m2/C2. Therefore,
q2 = (Fe r2) / k.
Substitute Fe = 2.04 × 10−5 N, r = 0.20 m, and k = 9 × 109:
q2 = (2.04 × 10−5 × (0.20)2) / (9 × 109).
(0.20)2 = 0.04. Thus,
q2 = (2.04 × 10−5 × 0.04) / (9 × 109) = (8.16 × 10−7) / (9 × 109).
= (8.16 / 9) × 10−7 − 9 = 0.907 × 10−16 → 9.07 × 10−17 (approximately).
Hence,
q ≈ √(9.07 × 10−17) = 3.01 × 10−9 C.
Step 7: Match q to the given form and find “a”
The question states that the charge on each sphere is given by
q = (a / 21) × 10−8 C.
We found q ≈ 3.0 × 10−9 C. Rewrite 3.0 × 10−9 C as 0.3 × 10−8 C. Thus,
0.3 × 10−8 = (a / 21) × 10−8.
So:
a / 21 = 0.3 ⟹ a = 0.3 × 21 = 6.3.
Final Answer
The value of “a” is 6.3.
