As a crude model for lightning, consider the ground to be one plate of a parallel-plate capacitor and a cloud at an altitude of 590 m to be the other plate. Assume the surface area of the cloud to be the same as the area of a square that is 0.50 km on a side.
(a) What is the capacitance of this capacitor?
(b) How much charge can the cloud hold before the dielectric strength of the air is exceeded and a spark (lightning) results?
for a I tried using c = Eo * A / d, but it didn't come out right. can someone help?
To find the capacitance of the parallel-plate capacitor, you need to use the formula:
C = ε₀ * (A / d)
Where:
C is the capacitance
ε₀ is the permittivity of free space (8.85 × 10⁻¹² F/m)
A is the surface area of one plate
d is the separation distance between the plates
In this case, the cloud is considered one plate of the capacitor, and the ground is the other plate. The given information about the cloud's surface area being the same as a square with 0.50 km on a side will be used.
(a) Calculating the capacitance:
To find the capacitance, you need to determine the surface area of the cloud and the separation distance between the plates.
The given information states that the surface area of the cloud is the same as a square with a side length of 0.50 km. Therefore, the surface area (A) can be calculated as:
A = (0.50 km)²
To convert the side length from kilometers to meters, you need to multiply by 1000:
A = (0.50 km)² * (1000 m / 1 km)²
Now you can calculate the surface area (A) in square meters.
Next, determine the separation distance (d). The separation distance is the altitude of the cloud from the ground, given as 590 m.
Now you can substitute the values into the capacitance formula:
C = ε₀ * (A / d)
Calculate the capacitance by plugging in the values for ε₀, A, and d.
(b) Calculating the maximum charge:
To determine the maximum charge the cloud can hold before sparking occurs, you need to use the formula:
Q = C * V
Where:
Q is the charge
C is the capacitance
V is the voltage at which the dielectric strength is exceeded
The dielectric strength of air is approximately 3 × 10⁶ V/m.
In this case, the voltage is unknown and needs to be determined. Since the voltage can be calculated using electric field and distance, we can use the formula:
V = E * d
Where:
V is the voltage
E is the electric field
d is the separation distance
The electric field (E) can be found using the formula:
E = V / d
Now you have all the information required to calculate the maximum charge (Q):
Q = C * V
Substitute the values of C and V into the equation to get the result.