Case I: \"Mini-lightning\" Suppose that after walking across a carpeted floor yo

Question

Case I: "Mini-lightning"
Suppose that after walking across a carpeted floor you reach for a doorknob, and just before you touch it a spark jumps 5 mm between your finger and the knob.

a.) Find the minimum voltage difference between your finger and the doorknob for this to occur.
V
(Does this seem large??)

b.) Let's explore further-- first, calculate the capacitance of this system. You may approximate the doorknob and your finger as identical circular parallel plates, each of average radius 1.4 cm. Then calculate the charge that is on your finger to cause this spark.
C = pF
Q = nC
(Does this seem large?)

Case II: Actual lightning
Estimate the ground to be one plate of a parallel-plate capacitor and a cloud at an altitude of 630 m to be the other plate of the same size. You may approximate them each as a square that is 0.45 km on a side.

a.) What is the capacitance of this capacitor?
nF

b.) Find the minimum voltage difference between cloud and ground needed for lightning to occur, and then find how much charge the cloud holds just before a lightning strike.
V
C

(How do the values in Part II compare to Part I?)

Explanation / Answer

1.

a)

Electric field strength of air is

E=3*10^6 N/C

Minimum Voltage difference

V=E*d =3*10^6*5*10^-3

V=15000 Vor 15 KV

b)

Area

A=pi*r^2=pi*(1.4*10^-2)^2

A=6.16*10^-4 m^2

Capaciance

C=eoA/d =(8.85*10^-12)*(6.16*10^-4)/(5*10^-3)

C=1.09 pF

c)

Charge

Q=CV =15000*1.09*10^-12

Q=16.35 nC

2.

a)

Area

A=(0.45*10^3)^2

A=202,500 m^2

C=eoA/d =(8.85*10^-12)*202500/630

C=2.84*10^-9 F

C=2.84 nF

b)

V=E*d=3*10^6*630

V=1.89*10^9 V

c)

Q=CV =2.84*10^-9*1.89*10^9

Q=5.38 C

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