Two identical small spheres of mass 2.0g are fastened to the ends of an insulating thread of length 0.60m The spheres are suspended by a hook in the ceiling from the centre of the thread. The spheres are given identical electric charges and hang in static equilibrium, with an angle of 30.0 degrees between the string halves. Calculate the magnitude of the charge on each sphere.
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To calculate the magnitude of the charge on each sphere, we can use the principle of electrostatics.
First, let's start by defining the variables given in the problem:
- Mass of each sphere: m = 2.0 g = 0.002 kg
- Length of the thread: L = 0.60 m
- Angle between the string halves: θ = 30.0 degrees
In static equilibrium, the gravitational force acting on each sphere is balanced by the electrostatic force. We can set up the equation as follows:
mg = k * Q^2 / r^2
where:
- m is the mass of each sphere
- g is the acceleration due to gravity (9.8 m/s^2)
- k is the electrostatic constant (8.99 x 10^9 Nm^2/C^2)
- Q is the charge on each sphere
- r is the distance between the spheres (half the length of the thread)
To find the distance (r), we can use trigonometry. Since the angle between the string halves is 30.0 degrees, the angle between the thread and the vertical direction is 15.0 degrees. Thus, r can be calculated as:
r = L / (2 * sin(15.0))
Now we can substitute the known values into the equation:
mg = k * Q^2 / r^2
Rewriting the equation to solve for the charge (Q):
Q^2 = (mg * r^2) / k
Q = sqrt((mg * r^2) / k)
Plugging in the given values:
m = 0.002 kg
g = 9.8 m/s^2
L = 0.60 m
k = 8.99 x 10^9 Nm^2/C^2
θ = 30.0 degrees
We can calculate r using the equation above, and then substitute the values into the final equation to find Q.