Nuclear fusion reactions require that positively charged nuclei be brought into close proximity, against the electrostatic repulsion. As a simple example, suppose a proton is fired at a second, stationary proton from a large distance away. What kinetic energy must be given to the moving proton to get it to come within 1.00e10-15 m of the target? Assume that there is a head-on collision and that the target is fixed in place.
To find the kinetic energy required to bring two positively charged protons within a distance of 1.00e-15 m, we can use the principles of conservation of mechanical energy and the concept of electrostatic potential energy.
Here's how you can calculate the answer:
Step 1: Determine the potential energy at the final separation.
The potential energy at a distance r for two charged particles with charges q1 and q2 is given by the formula:
PE = (k * q1 * q2) / r,
where k is Coulomb's constant (9.0 x 10^9 N⋅m²/C²).
In this case, q1 and q2 are both the charge of a single proton (+e = 1.6 x 10^(-19) C), and r is the final separation distance (1.00 x 10^(-15) m).
So, at the final separation:
PE_final = (9.0 x 10^9 N⋅m²/C²) * (1.6 x 10^(-19) C)² / (1.00 x 10^(-15) m)
Step 2: Calculate the initial potential energy.
At a large distance, the initial kinetic energy of the moving proton is entirely converted into potential energy when it reaches the final separation distance. So the initial potential energy (PE_initial) is zero.
Step 3: Use the conservation of mechanical energy.
According to the conservation of mechanical energy, the sum of the initial kinetic energy (KE_initial) and the initial potential energy (PE_initial) will be equal to the sum of the final kinetic energy (KE_final) and the final potential energy (PE_final).
Since PE_initial = 0, we have:
KE_initial + 0 = KE_final + PE_final
Step 4: Solve for the initial kinetic energy.
Rearrange the equation from step 3 and solve for KE_initial:
KE_initial = KE_final + PE_final
Step 5: Calculate the final kinetic energy.
At the final separation distance, the final potential energy is zero (PE_final = 0). Thus, the final kinetic energy is entirely due to the initial kinetic energy.
Therefore:
KE_final = KE_initial
Step 6: Find the kinetic energy required.
Calculate the initial potential energy (step 1) and the final kinetic energy (step 5) to find the required kinetic energy:
KE_required = KE_initial = KE_final
KE_required = (9.0 x 10^9 N⋅m²/C²) * (1.6 x 10^(-19) C)² / (1.00 x 10^(-15) m)
By plugging in the values and performing the calculation, you'll obtain the required kinetic energy in joules.