Example 17-2 Electric Potential Energy and Nuclear Fission When a nucleus of uranium-235 (92 protons and 143 neutrons) absorbs an additional neutron, it undergoes a process called nuclear fission in which it breaks into two smaller nuclei. One possible fission is for the uranium nucleus to divide into two palladium nuclei, each of which has 46 protons and is 5.9 * 10215 m in radius. The palladium nuclei then fly apart due to their electric repulsion. If we assume that the two palladium nuclei begin at rest and are just touching each other, what is their combined kinetic energy when they are very far apart? Set Up We can treat the two spherical nuclei as though they were point charges of q = +46e located at the centers of the two nuclei. (To motivate this, recall from Section 16-7 that the electric field outside a sphere of charge Q is the same as that of a point charge Q located at the sphere’s center.) Equation 17-3 then tells us the electric potential energy of the two palladium nuclei when they begin at rest. We’ll then use energy conservation to find the combined kinetic energy when the palladium nuclei are very far apart. Solve Use Equation 17-3 to solve for the initial electric potential energy when the two nuclei are just touching. Electric potential energy of two point charges: kq1q2 (17-3) Uelectric = r Palladium nucleus q = +46e Palladium nucleus q = +46e Conservation of mechanical energy: Ki + Uelectric, i = Kf + Uelectric, f (6-23) 2 × (5.9 × 10–15 m) equivalent to: q = +46e q = +46e 2 × (5.9 × 10–15 m) Each nucleus has charge +46e: q1 = q2 = +46(1.60 * 10219 C) = +7.36 * 10218 C The separation between the charges is twice the radius of either nucleus: r = 2(5.9 * 10215 m) The electric potential energy is Uelectric, i = 18.99 * 109 N # m2 >C 2 2 17.36 * 10-18 C2 2 kq1q2 = r 215.9 * 10-15 m2 = 4.1 * 10211 N · m = 4.1 * 10211 J Use energy conservation to find the combined final kinetic energy of the two palladium nuclei. The palladium nuclei begin at rest, so the initial kinetic energy is zero: Ki = 0 The palladium nuclei end up very far apart, so their separation is ­essentially infinite (r S ). The final electric potential energy is ­therefore zero: Uelectric, f = 0 From the energy conservation equation, the final kinetic energy is Kf = Ki + Uelectric,i 2 Uelectric,f = 0 + 4.1 * 10211 J 2 0 = 4.1 * 10211 J Reflect All of the initial electric potential energy is converted into kinetic energy of the palladium nuclei. The energy released by the fission of a single uranium-235 nucleus, 4.1 * 10211 J, is very small. But imagine that you could get 1.0 kg of uranium-235, which contains about 2.6 * 1024 uranium atoms, to undergo fission at once. (This is the principle of a fission bomb.) The released energy would be (2.6 * 1024) * (4.1 * 10211 J) = 1.1 * 1014 J, equivalent to the energy given off by exploding 26,000 tons of TNT. This gives an inkling of the terrifying amount of energy that can be released by a fission weapon.