The exponent of charge variables in physical law must be unity in order for them to apply to distributed charges

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The exponent of charge variables in physical law must be unity in order for them to apply to distributed charges

Let Z be a physical charge such as gravitational mass or electric charge for some physical body, that determines some variable E in the form
E = αZ^β

where α and β are parameters.
Suppose now that the body is divided up into m equal proportions. The amount per portion is then α(Z/m)^β and the total is
E_m = αZ^β/m^β-1

and therefore
limit_m→∞ E_m = 0 if β>1
=αZ if β=1
=sgn(α)*∞ if β<1

Thus only the case of β equal to 1 is meaningful physically.

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E = αZβ

Em = αZβ/mβ-1

limitm→∞ Em = 0 if β>1 =αZ if β=1 =sgn(α)*∞ if β<1

E = αZ^β

E_m = αZ^β/m^β-1

limit_m→∞ E_m = 0 if β>1
=αZ if β=1
=sgn(α)*∞ if β<1