# How high potential difference implies strong electric field around the conductors?

While reading capacitance I found out this statement that for large C, potential difference is small for a given charge means that a capacitor with large capacitance can hold large amount of charge at relatively small potential difference. "High potential difference implies strong electric field around the conductor". A strong electric field can ionise the surrounding air and discharge...

Please explain the lines in quotes maybe by physical interpretation or mathematically!

The key to understanding the quoted segment is understanding what electrical potential really means. The strict mathematical definition of the voltage, $$V$$, between any two points $$A$$ and $$B$$ in space is $$V=\oint_SE(r)\cdot ds$$, where $$S$$ is an arbitrary path between $$A$$ and $$B$$. As a result, voltage scales linearly with $$E(r)$$, which means that a higher voltage difference between two points also implies a stronger electric field between them.
Assume that the separation of the plates of the capacitor, $$d$$, is constant as is the electric field between the plates.
The magnitude of the electric field between the plates, $$E$$, is equal to the potential gradient, $$\frac Vd$$ where $$V$$ is the potential difference between the plates.