A metal sphere centered at the origin has a radius R and a net charge Q. The electric field at the point `x = 5 R` is `E _(0).` The sphere is replaced by a different metal sphere centered at the origin with radius 2R and a net charge `Q _(0),` The field at `x = 5R` is still `E _(0).` What is the ratio of `Q and Q _(0)` ?

Explanation

To solve the problem, we will apply Gauss's Law and the concept of electric fields due to spherical charge distributions.

### Step-by-Step Solution:

1. **Understanding the Electric Field Outside a Charged Sphere:**
For a charged metal sphere, the electric field outside the sphere (at a distance \( r \) from the center) is given by:
\[
E = \frac{kQ}{r^2}
...