Final answer:
The electric field's magnitude at the origin produced by a semi-circular arc with twice the charge of a quarter-circle arc would be twice the magnitude of the electric field due to the quarter-circle arc.
Step-by-step explanation:
The magnitude of the electric field at the origin produced by a semi-circular arc of charge can be found by applying principles of electrostatics. When charges are distributed over geometric shapes like arcs or rods, we use calculus to integrate and find the resultant electric field. However, if we are given that the electric field strength at the origin due to a quarter-circle of charge is E, then for a semi-circle which has twice the charge, the electric field strength would be 2E at the origin, assuming the radius of the arc is the same.
For example, if the electric field strength due to a quarter-circle arc with charge q is calculated and found to be E₁, then the electric field strength at the origin due to a semi-circular arc with twice the charge (2q) would simply be double that magnitude, which is 2E₁. This is because the electric field is a vector quantity and the field due to each infinitesimal part of the charge distribution adds up linearly.