Question

(II) An electric field greater than about $\mathbf{3}\times \mathbf{1}{\mathbf{0}}^{\mathbf{6}}\mathbf{V}\mathbf{/}\mathbf{m}$causes air to break down (electrons are removed from the atoms and then recombine, emitting light). See Section 17–2 andTable 17–3. If you shuffle along a carpet and then reach for a doorknob, a spark flies across a gap you estimate to be 1 mm between your finger and the doorknob. Estimate the voltage between your finger and the doorknob. Why is no harm done?

Short answer

The voltage between the finger and the doorknob is 3000 V. No harm is done because the amount of charge transferred is very small.

Step-by-step solution

Relation between Electric Field and Electric Potential

The electric potential energy per unit charge at any point in space is termed as the electric potential difference.

The potential difference $\left(V_{\mathrm{b}\mathrm{a}}\right)$between two points a and bseparated by a distance d apart in a region of uniform electric field is related to the electric field (E) by the following relation:

$E=-\frac{V_{ba}}{d}=-\frac{\mathrm{\Delta }V}{d}$ … (i)

Here, negative sign indicates that the direction of electric field is in the direction of decreasing potential.

Given information

The electric field generated between the finger and the doorknob is approximately, $E=3\times {10}^6\mathrm{V}/\mathrm{m}$

The distance between the finger and the doorknob is, $d=1\mathrm{m}\mathrm{m}=1\times {10}^{-3}\mathrm{m}$

Determination of the voltage between the finger and the doorknob

From equation (i), the expression for magnitude of electric field can be written as:

$E=\frac{\mathrm{\Delta }V}{d}$

So, the potential difference between finger and doorknob is,

$\begin{array}{rl}\mathrm{\Delta }V& =E.d\\ & =\left(3\times {10}^6\mathrm{V}/\mathrm{m}\right)\left(1\times {10}^{-3}\mathrm{m}\right)\\ & =3\times {10}^3\mathrm{V}\\ & =3000\mathrm{V}\end{array}$

Thus, the voltage between the finger and the doorknob is 3000 V.

The charge transferred between the finger and the doorknob is very small. Therefore, no harm is done.