Final answer:
To calculate the capacitance and potential difference of the thundercloud-earth system modeled as a parallel-plate capacitor, the area of the charge center and the distance to Earth's surface is used along with the permittivity of free space.
Step-by-step explanation:
The charge center of a thundercloud, which contains 20 C of negative charge and is located 3.0 km above Earth's surface, can be considered as one plate of a capacitor with Earth's surface as the other plate.
Given that the charge center has a radius of 1.0 km, we can use the formula for the capacitance of a parallel-plate capacitor to calculate the capacitance of the system:
C = ε0A/d
Where:
- C is the capacitance,
- ε0 is the permittivity of free space (ε0 = 8.85 x 10-12 F/m),
- A is the area of the plates, and
- d is the separation between the plates.
For a radius r of 1.0 km, the area A is πr2 = π3.14 x 106 m2. The separation d is 3.0 km = 3000 m.
The potential difference V can then be found from the relationship Q = CV, where Q is the charge on the plates (20 C).
The capacitance C is therefore calculated as:
C = (8.85 x 10-12 F/m)(π3.14 x 106 m2)/3000 m = 9.25 x 10-9 F
The potential difference V is given by:
V = Q/C = 20 C / 9.25 x 10-9 F = 2.162 x 109 V