@ZapperZ:
But, the order of magnitude is not what the OP is concerned in this particular case. the rest energy of an electron is 0.511 MeV, thus the minimal energy for producing a pair is 1.022 MeV.
@OP:
Saying the value of the potential at a point has a particular value does not make any sense until you define a reference point where the potential is known. Consequently, what is physical is potential difference.
My thoughts:
I would say that electric fields are more important than potential differences. Namely, suppose a virtual pair is produced, with each particle traveling at speed v. We have borrowed an energy:
<br />
\Delta E = \frac{2 \, m \, c^{2}}{\sqrt{1 - v^{2}/c^{2}}}<br />
and we had better return it in time of the order:
<br />
\Delta t \sim \hbar / \Delta E<br />
In this time (ignoring acceleration of the charged particles for simplicity), the two particles will cover a distance)
<br />
\Delta x = v \, \mathrm{cos}{\theta} \, \Delta t<br />
If the positron moves along the field and the electron opposite of the field, then the electric forces would gain energy:
<br />
2 \, e \, K \, \Delta x<br />
If this energy is greater than or equal to the borrowed energy, pair creation might become possible. Combining everything together, we get:
<br />
2 \, e \, K \, v \, \frac{\hbar}{\Delta E} \gtrsim \Delta E<br />
<br />
K \gtrsim \frac{(\Delta E)^{2}}{2 \, e \, \hbar \, v} = \frac{2 (m c^{2})^{2}}{e \, \hbar \, c} \, \frac{1}{v/c (1 - v^{2}/c^{2})}<br />
The function
<br />
\frac{1}{x(1 - x^{2})}<br />
has a local minimum for x_{0} = 1/\sqrt{3} which is 3 \, \sqrt{3}/2. Thus, a minimal electric field that would create pair production is:
<br />
K_{\mathrm{min}} = 3 \, \sqrt{3} \, \frac{(m c^{2})^{2}}{e \, \hbar \, c} = 3 \times 1.73 \, \frac{(0.511 \, \mathrm{MeV})^{2}}{1 \, \mathrm{e} \times 197.4 \, \mathrm{MeV} \cdot \mathrm{fm}} = 6.9 \times 10^{-3} \, \frac{\mathrm{MV}}{\mathrm{fm}} = 6.9 \times 10^{18} \, \frac{\mathrm{V}}{\mathrm{m}}<br />
This field is very strong! In fact, it would take a distance of only 0.14 pm to generate a 1 MV potential difference.
