a circuit consists of two identical parallel metallic plates connected by identical metallic springs to a 102V battery. With the switch open, the plates are uncharged, are separated by a distance d=7.80mm, and have a capacitance C=2.00microF. When the switch is closed, the distance between the plates decreases by a factor of .500. (a)How much charge collects on each plate
(b)What is the spring constant for each spring?
To solve this problem, we need to consider the principles of capacitance and the behavior of springs in a parallel plate capacitor circuit.
(a) To find the charge on each plate, we can use the formula:
Q = C * V
where Q is the charge, C is the capacitance, and V is the voltage.
In this case, the capacitance C is given as 2.00 microFarads (μF) and the voltage V is given as 102V. Plugging in these values, we get:
Q = (2.00 μF) * (102V)
Now we can calculate the charge Q:
Q = 204 μC
Therefore, each plate accumulates a charge of 204 microCoulombs.
(b) Now let's move on to finding the spring constant for each spring. In a parallel plate capacitor circuit, when the distance between the plates changes, the springs connecting the plates provide the restoring force.
The spring constant (k) can be calculated using Hooke's law formula:
F = -k * x
where F is the force, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, the distance between the plates decreases by a factor of 0.500. Considering the equilibrium condition, the force F would be equal to the electrostatic force between the charged plates.
By equating these two forces, we can write:
k * x = Q^2 / (2 * C)
Here, x denotes the displacement, which equals the original distance between the plates (7.80 mm) minus the final distance after closing the switch (0.500 * 7.80 mm).
Plugging in the values, we get:
k * (7.80 mm - 0.500 * 7.80 mm) = (204 μC)^2 / (2 * 2.00 μF)
Simplifying the equation and solving for k, we find:
k = [(204 μC)^2 / (2 * 2.00 μF)] / [0.500 * 7.80 mm]
After performing the calculations, the spring constant for each spring will be determined.