Charge on fingertip from electrostatic spark to metal truck

[Josh123]

here's a problem that I am currently working on. I know the theory, but I'm not quite sure how to start this particular problem:

"Thomas was crawling around on the rug. When he reached for a metal truck, a prominent spark lasting 5mmsec appeared between his fingertip and the object. His fingertip was about 2 mm from his toy. His finger burnt (the area of the burned region was of 10^-4 m^2)

On that day, the air was cold and dry causing it to become conducting when the electric field reached 3*10^6 N/C."

My question is, how you I determine the charge on the child's fingertip?
How do I estimate the resistance of the dry air between the toy
truck and the child's fingertip? (I just would like to know how to start this problem.. you don't have to do the entire thing)

Thank you in advance

 

[Pieter Kuiper]

First you determine the voltage that gives a 2 mm spark (the breakdown field strength is given, but in other units than V/m).

Then you equate the energy when a charge Q goes through the calculated potential difference with the energy needed to produce such a burn (I do not know how to estimate that - are there more things given?).

 

[thepatientmental]

To start this problem, we can use the formula for electric field:

E = V/d

Where E is the electric field, V is the potential difference, and d is the distance between the two objects. In this case, the potential difference can be calculated as the work done to move one unit of charge from one object to the other. We can estimate this value by using the formula:

V = W/Q

Where V is the potential difference, W is the work done, and Q is the charge on the child's fingertip.

To determine the charge on the child's fingertip, we can rearrange the equation to solve for Q:

Q = W/V

We can estimate the work done by the spark by using the formula for electrical energy:

W = QV

Since we know the potential difference (V) and the time (5mmsec) for which the spark lasted, we can estimate the work done.

Next, we can use the formula for resistance:

R = ρL/A

Where R is the resistance, ρ is the resistivity of the material (in this case, the dry air), L is the length of the air gap (2 mm in this case), and A is the cross-sectional area (10^-4 m^2 in this case).

To estimate the resistivity of dry air, we can use the formula:

ρ = 1/σ

Where ρ is the resistivity and σ is the conductivity of air. The conductivity of air can be determined using the formula:

σ = ne^2τ/m

Where n is the number of free electrons, e is the charge of an electron, τ is the relaxation time, and m is the mass of an electron. All of these values can be found in a table or can be estimated.

Once we have estimated the resistance of the dry air, we can use Ohm's law to calculate the current flowing through the air gap:

I = V/R

From there, we can use the formula for charge:

Q = It

Where Q is the charge, I is the current, and t is the time (5mmsec in this case).

I hope this helps you get started on the problem. Remember to always use the appropriate formulas and units, and to estimate values if necessary. Good luck!