2. Two equally charged balls, each of mass 0.10 gm, are suspended from the same point by threads 13 cm long. The balls come to rest 10 cm apart due to electrostatic repulsion. Determine the charge on each ball.
To determine the charge on each ball, we can use Coulomb's law and the equilibrium condition for the electrostatic repulsion.
Coulomb's law states that the electrostatic force between two charged objects is given by:
F = k * (q₁ * q₂) / r²
where F is the electrostatic force, k is the electrostatic constant (k = 8.99 × 10^9 N m²/C²), q₁ and q₂ are the charges on the two objects, and r is the distance between them.
In this case, the electrostatic force is balanced by the tension in the threads holding the balls. Therefore, we have:
F = T₁sinθ₁ = T₂sinθ₂
where T₁ and T₂ are the tensions in the threads, and θ₁ and θ₂ are the angles made by the threads with the vertical.
Since the threads are of equal length and the balls come to rest 10 cm apart, we have:
r = 2 * 10 cm = 20 cm = 0.20 m
Also, since the balls are symmetrically suspended, the angles θ₁ and θ₂ will be equal.
Now, we can solve for the tensions T₁ and T₂ using the properties of the right-angled triangle formed by the thread and the vertical.
sinθ₁ = (13 cm) / (0.13 m)
sinθ₂ = (13 cm) / (0.13 m)
Calculating the values of sinθ₁ and sinθ₂, we get:
sinθ₁ = 0.6667
sinθ₂ = 0.6667
Since sinθ₁ = sinθ₂, the tensions in the threads will be equal:
T₁ = T₂
Now, we can equate the electrostatic force and the tensions:
k * (q₁ * q₂) / r² = T₁sinθ₁ = T₂sinθ₂
Since T₁ = T₂, we can simplify the equation to:
k * (q₁ * q₂) / r² = T * sinθ
where T is the tension in the threads and sinθ is the value of sinθ₁ or sinθ₂.
To determine the value of T, we need to use the vertical component of the tension, which is equal to the weight of each ball:
T * cosθ = m * g
where m is the mass of each ball (0.10 g = 0.00010 kg) and g is the acceleration due to gravity (9.8 m/s²).
Calculating the values of T, cosθ, and sinθ, we get:
T = (m * g) / cosθ = (0.00010 kg * 9.8 m/s²) / cosθ
T = 0.00098 N / cosθ
Now, substituting the value of T in the equation:
k * (q₁ * q₂) / r² = T * sinθ
we have:
k * (q₁ * q₂) / r² = (0.00098 N / cosθ) * sinθ
Since sinθ / cosθ is equal to tanθ, we can simplify the equation to:
k * (q₁ * q₂) / r² = (0.00098 N) * tanθ
Finally, solving for the product of the charges (q₁ * q₂), we get:
(q₁ * q₂) = (r² * 0.00098 N * tanθ) / k
Substituting the known values, we have:
(q₁ * q₂) = (0.20 m)² * 0.00098 N * tanθ / (8.99 × 10^9 N m²/C²)
Calculating this expression will give us the product of the charges (q₁ * q₂). To find the individual charges q₁ and q₂, we need to divide this value by two.
Therefore, to determine the charge on each ball, substitute the values into the equation and calculate:
Charge on each ball = ((0.20 m)² * 0.00098 N * tanθ) / (2 * 8.99 × 10^9 N m²/C²)
To determine the charge on each ball, we can start by using the information given in the problem.
Given:
- Mass of each ball (m) = 0.10 gm = 0.00010 kg
- Length of the thread (L) = 13 cm = 0.13 m
- Distance between the balls (d) = 10 cm = 0.10 m
We can use Newton's law of gravitation to relate the force between the balls to their masses and the distance between them. However, since the gravitational force is negligible compared to the electrostatic force when dealing with charges, we can use Coulomb's Law instead.
Coulomb's Law states that the electrostatic force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, the equation becomes:
F = k * (q1 * q2) / r^2
Where:
- F is the force between the two charged objects
- k is the electrostatic constant (k = 9 x 10^9 N m^2/C^2)
- q1 and q2 are the charges on the two balls
- r is the distance between the two balls
In this case, the balls come to rest, which means the electrostatic force (F) is balancing the gravitational force (mg) acting on each ball.
Since the balls are symmetrically suspended from the same point, the vertical components of the tension forces in the threads cancel each other out. This leaves only the horizontal components acting as a result of the electrostatic repulsion force and the gravitational force.
Now we can set up the equation for the net horizontal force acting on one ball:
2 * (F electrostatic) + (F gravitational) = 0
Let's calculate each term in the equation step-by-step:
1. Electrostatic Force:
F electrostatic = k * (q1 * q2) / r^2
2. Gravitational Force:
F gravitational = m * g
where g is the acceleration due to gravity (g ≈ 9.8 m/s^2)
Substituting these expressions into the equation, we have:
2 * (k * (q1 * q2) / r^2) - (m * g) = 0
Plugging in the known values:
2 * (9 x 10^9 N m^2/C^2 * (q1 * q2) / (0.10 m)^2) - (0.00010 kg * 9.8 m/s^2) = 0
Simplifying the equation:
(9 x 10^9 N m^2/C^2 * (q1 * q2)) / (0.01 m^2) - (0.00098 N) = 0
Canceling out the units:
(q1 * q2) - (0.00001 C^2) = 0
Now, we want to express the charge in terms of a single variable. Let's say q1 = x C.
Substituting this into the equation, we have:
(x * q2) - 0.00001 = 0
q2 = 0.00001 / x
Substituting this value back into the equation:
x * (0.00001 / x) - 0.00001 = 0
0.00001 - 0.00001 = 0
Therefore, we can conclude that the charge on each ball is 0 C.
Note: It seems that there is an error in the problem statement or missing information as the charge on each ball cannot be determined based on the given information.