Two small aluminum spheres, each having mass 2.65×10−2 , are separated by 85.0 .

How many electrons would have to be removed from one sphere and added to the other to cause an attractive force between the spheres of magnitude 4.00×104 (roughly four ton)? Assume that the spheres may be treated as point charges.

To find the number of electrons that need to be transferred, we need to calculate the charge on each sphere first.

The charge on an object can be expressed as: q = n * e,

where q is the total charge, n is the number of excess or deficit electrons, and e is the elementary charge, approximately 1.6 x 10^-19 C.

First, we calculate the charge required to generate the attractive force of magnitude 4.00 × 10^4 N between the spheres. Since the force is attractive, the charges on the spheres will have opposite signs.

Let's assume that the first sphere needs to lose \(n_1\) electrons and the second sphere needs to gain \(n_2\) electrons.

The charge on the first sphere will be q1 = -\(n_1\) * e (negative because it loses electrons), and the charge on the second sphere will be q2 = \(n_2\) * e (positive because it gains electrons).

Now, we can use Coulomb's Law to relate the force and distance to the charges:

F = k * |q1 * q2| / r^2,

where F is the magnitude of the force, k is the electrostatic constant (9.0 × 10^9 N·m^2/C^2), q1 and q2 are the charges on the spheres, and r is the separation distance between the spheres.

Plugging in the given values, we have:

4.00 × 10^4 N = (9.0 × 10^9 N·m^2/C^2) * |(-\(n_1\) * e) * (\(n_2\) * e)| / (85.0 m)^2.

Next, we can simplify the equation:

4.00 × 10^4 N = (9.0 × 10^9 N·m^2/C^2) * |n_1 * n_2 * e^2| / 85.0^2 m^2.

Simplifying further, we can cancel out the constants:

4.00 × 10^4 = |n_1 * n_2| * (9.0 × 10^9 / 85.0^2) * e^2.

Finally, we can solve for \(n_1\) * \(n_2\):

|n_1 * n_2| = (4.00 × 10^4 / (9.0 × 10^9 / 85.0^2)) * (1 / e^2).

Now, let's calculate the value of |n_1 * n_2|:

|n_1 * n_2| = (4.00 × 10^4 / (9.0 × 10^9 / 85.0^2)) * (1 / (1.6 × 10^-19 C)^2).

|n_1 * n_2| ≈ 2.02 × 10^27.

Since the number of electrons is always a whole number, we can assume that both \(n_1\) and \(n_2\) need to be approximately the square root of 2.02 × 10^27.

\(n_1\) ≈ \(n_2\) ≈ √(2.02 × 10^27).

Therefore, the number of electrons that need to be transferred from one sphere to the other is approximately √(2.02 × 10^27).