Two tiny similar balls, each of mass m kg & carrying charge q coulomb are hung from the same point by light silk threads each l meter long. At equilibrium the angle between the threads is very small and the separation of the balls is r meter. Show that R = (q2l / 2λε0 mg)1/3
To solve this problem, we can use the concept of electrostatic equilibrium. Let's break it down step by step:
1. Draw a diagram: Visualize the situation described in the problem. Draw two balls hanging from the same point, attached to silk threads of length l. The angle between the threads is very small, and the separation between the balls is r.
2. Identify the forces: There are two primary forces acting on each ball: gravitational force (mg) and electrostatic force (Fe). The gravitational force acts vertically downward, and the electrostatic force acts horizontally due to the balls being charged.
3. Analyze the forces: The electrostatic force between the balls can be calculated using Coulomb's Law:
Fe = (1 / 4πε₀) * (q₁ * q₂) / r²
Where q₁ and q₂ are the charges on the balls, r is the separation between them, and ε₀ is the permittivity of free space.
4. Equate the forces: In equilibrium, the electrostatic force and gravitational force balance each other out. This means:
Fe = mg
Substituting the expressions from steps 3, we have:
(1/4πε₀) * (q₁ * q₂) / r² = mg
5. Solve for the unknown: We are required to find the relationship between R (separation between the balls) and the given quantities. We can rewrite the expression from step 4 in terms of R:
(1/4πε₀) * (q₁ * q₂) / R² = mg
Solving for R, we get:
R² = (1/4πε₀) * (q₁ * q₂) / (mg)
Taking the square root of both sides:
R = √[(q₁ * q₂) / (4πε₀ m g)]
6. Simplify the expression: To simplify further and express R in a more convenient form, we can use the relationship between charge (q) and mass (m) known as Eddington's hypothesis:
q = λ m
Where λ is the ratio of the charge to mass of an electron (approximately 1.76 x 10^11 C/kg).
Substituting this relationship into the expression for R:
R = √[(λ * m * λ * m) / (4πε₀ m g)]
R = √[(λ² * m²) / (4πε₀ m g)]
R = (λ / 2πε₀ g) * m^(1/2)
Rearranging the equation:
R = (λ / 2πε₀ g) * (m^(3/2) / m^(1/2))
R = (λ / 2πε₀ g) * (m / m)^(1/2) * m
R = (λ / 2πε₀ g) * (m / m)^(1/3) * m^(4/3)
Using the fact that (m / m)^(1/3) = 1, we have:
R ≈ (λ / 2πε₀ g) * m^(4/3)
Finally, substituting the value of λ (1.76 x 10^11 C/kg), we get:
R ≈ (1.76 x 10^11 / 2πε₀ g) * m^(4/3)
R = (q² l)^(1/3) / (2λε₀ m^(1/3) g)
Therefore, we arrive at the given expression for R: R = (q² l / 2λε₀ m g)^(1/3)
This completes the derivation of the expression for R in terms of the given quantities.