Paschen's Law - Deepakkumar Yadav

Paschens Law

Q. State and explain Paschens Law ?

It States that, ” Breakdown voltage of uniform field gap is unique function of product of p, i.e. electrode gap distance for a particular gas and for a given electrode material.

According to Townsend’s breakdown Criteria y and α both are functions of E/P :

y (exp (α – n) d – 1) = 1… (1)

Let α / p = f₁(E / P) and … (2)

y = f₂(E / P … (3)

Also E = V / d … (4)

Substituting for E in the expression for y and α,

(α / α – n) = (exp (α – n) d – 1) = 1 … (5)

Substituting for E in Equation (5) for α and y and rewriting, we get,

F₂(V pd) [exp {pd f₁(V / pd)} – 1] = 1 …(6)

Figure B

This equation shows relationship between V and pd and implies that the breakdown voltage varies as product pd varies.
Knowing nature of functions F₁and F₂, we can say

V = f (pd) …(7)

This equation is also known as Paschen’s Law.
This prove that a breakdown voltage of a uniform field gap is a unique function of the product of the gas pressure (p) and electrode gap (d) for a particular gas and for a given electrode material.
Now, from Paschen’s Law, we can obtain the breakdown voltage of a spark gap by rewriting the breakdown condition and substituting values for α and y in terms of pd product.

Thus, we get,

d = 1 / α [ ln (1+ 1 / y)]

= 1 / P f₁(v / pd) ln [1 + 1 / f₂(v / pd)]

Where, α / p = f₁(E / P) and f₂(E / P)

f₁and f₂are same functions

E = V / d

α may be assumed to follow an exponential function

∴ α = A_{pe- Bp/ E}= A_{pe- Bpd/ v}

Substituting by α only

d = 1 / A_pe^{– Bpd/ v}ln [1 + 1 / y]

Thus minimum values for V can be obtained by making dv / d (pd) = 0, which gives rise to,

pd_min= e / A ln [1 + 1/y]

where, e = 2.178

V_min= e B / A ln [1 + 1/y]

In order to account for the effect of temperature, Paschen’s law is generally stated as V = f (Nd)

Where, N – density of gas molecules

As the pressure of gas changes with temperature as per the gas law pv = NRT

Where, N – volume of gas

T – temperature

R – constant

It is essential to consider N,

Based on experimental results, the breakdown potential of air is expressed as a power function in pd as,

V = 24.22 (293 pd / 760 T) + (293 pd / 760 T)^1/2

At 760 torr and 293 K

E = V / d = 24.22 + 6.08 / √d kV / cm

A point worth nothing here is that the breakdown voltage at constant pressure and temperature (20^oc) and

atomic pressure (760 torr) as 30 kV / cm for (293pd / 760 T) = 1.

Limitations of Paschen’s Law