On a dry winter day, if you scuff your feet across a carpet, you build up a char

Question

On a dry winter day, if you scuff your feet
across a carpet, you build up a charge and get
a shock when you touch a metal doorknob.
In a dark room you can actually see a spark
about 2 cm long. Air breaks down at a field
strength of 3 × 106 N/C.
Assume that just before the spark occurs,
all the charge is in your finger, drawn there by
induction due to the proximity of the door-
knob. Approximate your fingertip as a sphere
of diameter 1.44 cm, and assume that there is
an equal amount of charge on the doorknob
2 cm away.
How much charge have you built up?
Answer in units of C.

Explanation / Answer

Let : x = 2 cm, Ebreak = 3 × 106 N/C, and r = 0.72 cm = 0.0072 m. The electric field of a point charge is E = ke(q/r^2) . In order for air to break down, there must be a field strength of Ebreakdown just outside your fingertip, or Efinger = Ebreakdown = ke(q/r^2 ). Thus q =Ebreakdown r^2/ke =(3 × 106 N/C) (0.0072 m)2 8.98755 × 109 N · m2/C2 = 1.73039 × 108 C .

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