Consider an attractive nuclear force acting between two protons that can be expressed as F = Ce^ -kr /r^2 . Assume that k = 1 fermi ⁻¹ and the repulsive electric force between the protons is exactly balanced by the attractive nuclear force at a separation of 5 fermi . Determine the value of C .

Step-by-step solution

The repulsive electric force between two protons is given by Coulomb's law: F_ e = 1 4 ₀ e^2 r^2 The attractive nuclear force is given by: F_ n = C e^ -kr r^2 At r = 5 fermi , these forces balance each other, so F_ e = F_ n . 1 4 ₀ e^2 r^2 = C e^ -kr r^2 C = e^2 4 ₀ e^ kr Given k = 1 fermi ⁻¹ and r = 5 fermi , we have kr = 5 . Substituting the known values: 1 4 ₀ = 9 10^9 N m ^2/ C ^2 e = 1.6 10⁻¹⁹ C e^5 148.4 C = (9 10^9) (1.6 10⁻¹⁹)^2 e^5 C = 9 10^9 2.56 10⁻³⁸ 148.4 C = 23.04 10⁻²⁹ 148.4 C 3419 10⁻²⁹ N m ^2 C 3.4