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 #
