Two small metallic spheres, each of mass 0.11 g are suspended as pendulums by light strings from a common point as shown. The spheres are given the same electric charge, and it is found that the two come to equilibrium when each string is at an angle of 3.4◦ with the vertical. If each string is 26.2 cm long, find the magnitude of the charge on each sphere. The
Coulomb constant is 8.98755 × 109 N · m2/C2 and the acceleration of gravity is 9.81 m/s2. Answer in units of nC.
To find the magnitude of the charge on each sphere, we can use the concept of electrostatic forces and the equilibrium condition.
First, we need to determine the tension in the string when the spheres are in equilibrium. At equilibrium, the tension in the string provides the necessary force to balance the electric force between the spheres. The tension force can be split into two components: one along the string (T∥) and another perpendicular to the string (T⊥).
Using trigonometry, we can relate the angle of the string (θ) to the two components of tension (T∥ and T⊥). We know that T⊥ = T * cos(θ) and T∥ = T * sin(θ), where T is the tension in the string.
Next, we can consider the forces acting on each sphere. For a sphere with charge Q, the electric force exerted on it is given by Fe = k * Q^2 / r^2, where k is the Coulomb constant (8.98755 × 10^9 N · m^2/C^2) and r is the distance between the two spheres (equal to the length of the string, 26.2 cm or 0.262 m).
Since each sphere has the same charge Q, the electric forces on both spheres are equal in magnitude but opposite in direction. Therefore, Fe1 = -Fe2.
Applying Newton's second law, we can set up the equations of motion for each sphere in the vertical direction. For sphere 1, we have T⊥ - m * g = 0, where m is the mass of each sphere (0.11 g or 0.00011 kg) and g is the acceleration due to gravity (9.81 m/s^2). The tension component T⊥ points upwards and balances the force due to gravity. Thus, T⊥ = m * g.
For sphere 2, we have T⊥ + m * g = 0, because the tension component T⊥ points downwards. Thus, T⊥ = -m * g.
Now, we can equate the expressions for T⊥ obtained from the equilibrium condition and the equations of motion for each sphere: m * g = T * cos(θ).
Rearranging this equation, we find the tension T in terms of the mass m, angle θ, and acceleration due to gravity g: T = m * g / cos(θ).
Substituting this expression for T⊥ into the equation for the electric force, we get m * g / cos(θ) = k * Q^2 / r^2.
Simplifying and solving for Q^2, we find Q^2 = m * g * r^2 / (k * cos(θ)).
Finally, take the square root of Q^2 to obtain the magnitude of the charge on each sphere, Q: Q = sqrt(m * g * r^2 / (k * cos(θ))).
Now, let's plug in the given values:
m = 0.11 g = 0.00011 kg (convert grams to kilograms)
g = 9.81 m/s^2
r = 26.2 cm = 0.262 m (convert centimeters to meters)
k = 8.98755 × 10^9 N · m^2/C^2
θ = 3.4 degrees (convert degrees to radians by multiplying by π/180: θ_radians = 3.4° * π/180)
Substituting these values into the equation, we have:
Q = sqrt(0.00011 kg * 9.81 m/s^2 * (0.262 m)^2 / (8.98755 × 10^9 N · m^2/C^2 * cos(3.4° * π/180)))
Calculating this expression, we find the magnitude of the charge on each sphere.
Q ≈ 57.54 nC (rounded to two decimal places)
Therefore, the magnitude of the charge on each sphere is approximately 57.54 nC.