Let two point charges #q_1andq_2# be located in air at a distance #r# from each other.

Magnitude of Force between the two charges is given by Coulomb's Law

#|vecF_(12 " air")|= k_e (q_1 q_2)/ r^2# ......(1)

where #k_e=1/(4piepsilon_0)# is Coulomb's constant# = 8.99×10^9 N m^2 C^-2#, and the constant #ε_0# is the permittivity of free space.

When the same charges are placed in a medium of relative permittivity #epsilon_r#, the electric field between the charges is decreased relative to vacuum by this value. Hence, we have the relation for force as

#|vecF_(12 " med")|= k_e/epsilon_r (q_1 q_2)/ r_"med"^2# ......(2)

To meet the given condtion equating (1) and (2) we get

#k_e (q_1 q_2)/ r^2=k_e/epsilon_r (q_1 q_2)/ r_"med"^2#

#=> r_"med"^2=r^2/epsilon_r #

#=> r_"med"=r/sqrtepsilon_r #