The force between two charged particles is given by Coulomb's law, which states that the force (F) between two charged particles is directly proportional to the product of their charges (q1 and q2) and inversely proportional to the square of the distance (r) between them:
F = k * (q1 * q2) / r^2
Where: F = Force between the particles k = Coulomb's constant (a constant value) q1, q2 = Charges of the particles r = Distance between the particles
In this case, an alpha particle has twice the charge of a proton. Let's assume the charge of a proton is q, so the charge of an alpha particle is 2q.
Given that two protons separated by a distance 'd' exert a force 'F' on each other, we can set up the equation:
F = k * (q * q) / d^2
To find the separation between the alpha particles (d') so that they also exert a force 'F' on each other, we need to equate the forces:
F = k * ((2q) * (2q)) / d'^2
Since both forces are equal to 'F', we can set up an equation to find d':
k * (q * q) / d^2 = k * ((2q) * (2q)) / d'^2
Cancelling out the constants and rearranging the equation, we get:
(q * q) / d^2 = ((2q) * (2q)) / d'^2
Simplifying further:
d'^2 = (d^2) / (4)
Taking the square root of both sides:
d' = d / 2
Therefore, the separation between the alpha particles (d') must be half the separation between the protons (d) in order for them to exert the same force on each other.