Lighting strike: Storm clouds build up large negative charges that dwell in a charge center. Suppose a storm cloud has -25 C in a charge center located 10 km above the ground. The negative charge center attracts an equal amount of positive charge that is spread on the ground below the cloud. The charge center and the ground function as a charged capacitor, with a potential difference of 4 x 10^8 V. The large electric field between these two electrodes may ionize the air leading to a conducting path between the cloud and the ground.
a) What is the magnitude of the electric field between the charge center and the ground?
b) What is the capacitance of the charge center ground system?
c) If the cloud transfers all of its charge to the ground via several rapid lighting flashes lasting a total of 1 s what is the average power.
a) To find the magnitude of the electric field between the charge center and the ground, we can use the formula:
E = V / d
Where:
E is the electric field,
V is the potential difference, and
d is the distance between the charge center and the ground.
In this case, the potential difference is given as 4 x 10^8 V and the distance is 10 km (which is equal to 10,000 m). Plugging these values into the formula, we get:
E = (4 x 10^8 V) / (10,000 m)
Simplifying, we find:
E = 4 x 10^4 V/m
So, the magnitude of the electric field between the charge center and the ground is 4 x 10^4 V/m.
b) The capacitance of the charge center ground system can be calculated using the formula:
C = Q / V
Where:
C is the capacitance,
Q is the charge stored in the capacitor, and
V is the potential difference.
In this case, the charge stored in the capacitor is given as -25 C (since the storm cloud has a negative charge) and the potential difference is given as 4 x 10^8 V. Plugging these values into the formula, we have:
C = (-25 C) / (4 x 10^8 V)
Simplifying, we find:
C = -6.25 x 10^-8 F
So, the capacitance of the charge center ground system is -6.25 x 10^-8 Farads.
Note: The negative sign in the capacitance indicates that the charge center and the ground have opposite charges.
c) To find the average power, we need to know the total charge transferred and the time duration. The average power can be calculated using the formula:
P = W / t
Where:
P is the average power,
W is the work done, and
t is the time duration.
In this case, the total charge transferred is -25 C (negative charge value) and the time duration is given as 1 s. Plugging these values into the formula, we get:
P = (-25 C) / (1 s)
Simplifying, we find:
P = -25 W
So, the average power of the lightning flashes is -25 watts.