To solve this problem, we can use the formula for the electric field due to a uniformly charged sphere:
E = (k * Q) / r^2,
Where:
E is the electric field,
k is the electrostatic constant (k = 9 × 10^9 N·m^2/C^2),
Q is the total charge of the sphere,
and r is the radius of the sphere.
Given that the original field magnitude is e0 and the final field magnitude is 2e0, we can set up the following equation:
e0 = (k * Q_initial) / (r^2),
2e0 = (k * Q_final) / (r^2).
By dividing these two equations, we can eliminate k and r^2:
(e0 / 2e0) = (Q_initial / Q_final).
Simplifying the equation gives us:
1/2 = Q_initial / Q_final.
Now, we can solve for Q_initial by substituting Q_final = q + Q_initial, where q is the charge added to the surface:
1/2 = Q_initial / (q + Q_initial).
Cross-multiplying, we get:
2 * Q_initial = q + Q_initial.
Simplifying this equation gives:
Q_initial = q.
Therefore, the amount of charge added to the surface is equal to the value of the initial charge, Q_initial, which is q.