Two identical copper spheres are separated by 1 meter in vacuum . How many electrons would have to be removed from one sphere and added to other sphere so that they now attract each other with a force 0.9 Newton?
To find out how many electrons need to be transferred between the two copper spheres, we need to determine the electric charge on each sphere first.
The force of attraction between two charged objects can be calculated using Coulomb's Law:
F = k * (q1 * q2) / r^2
Where:
F is the force of attraction
k is the electrostatic constant (k = 8.99 x 10^9 Nm^2/C^2)
q1 and q2 are the electric charges on the two spheres
r is the distance between the centers of the spheres
Since the spheres are initially neutral, each sphere begins with an equal number of electrons and protons. Copper has 29 protons per atom, so each sphere starts with the charge from these protons.
The charge on one electron is approximately -1.6 x 10^-19 coulombs. To calculate the initial charge on each sphere, we need to find the number of electrons in one sphere.
The volume of a sphere is given by the formula:
V = (4/3) * π * r^3
Since the spheres are identical, their volumes are the same, so we can equate the volumes:
(4/3) * π * r^3 = (4/3) * π * r^3
Simplifying the equation, we have:
r^3 = r^3
Now, substitute the given distance between the spheres, 1 meter:
1^3 = 1^3
1 = 1
The radius of each sphere is the same, so we can consider the radius of one sphere as r and rewrite the equation as:
r = r
Now we know that the distance between the centers of the spheres is equal to the radius of each sphere, r.
To calculate the initial charge on one sphere, we need to find the number of electrons, n, in that sphere. We can use the density of copper, ρ = 8.96 g/cm^3, and the mass of one copper atom to find the number of atoms and then multiply by the number of electrons per atom.
The atomic mass of copper is approximately 63.55 g/mol. The Avogadro's number, N_A, tells us the number of atoms in one mole of a substance, which is approximately 6.022 x 10^23. Therefore, the mass of one copper atom is given by:
m_atom = (63.55 g/mol) / (6.022 x 10^23 atoms/mol)
Now, we can calculate the number of atoms in one sphere using its mass:
m_sphere = ρ * V_sphere
m_atom * n = m_sphere
Substituting the volume equation (4/3) * π * r^3 for V_sphere:
m_atom * n = ρ * (4/3) * π * r^3
We have previously established that the distance between the spheres is equal to the radius, r. So, substituting r:
m_atom * n = ρ * (4/3) * π * (1 meter)^3
Simplifying, we have:
m_atom * n = ρ * (4/3) * π
Now, let's calculate the initial charge on one sphere using the number of electrons:
q_initial = n * (-1.6 x 10^-19 coulombs)
Now that we have the initial charges on each sphere, we can use Coulomb's Law to find out the number of electrons that need to be transferred to create a force of 0.9 Newton:
0.9 N = (8.99 x 10^9 Nm^2/C^2) * (q_initial * q_final) / (1 meter)^2
Rearranging the equation to solve for q_final:
q_final = (0.9 N * (1 meter)^2) / (8.99 x 10^9 Nm^2/C^2)
Finally, divide q_final by the charge on one electron to find the number of electrons to be transferred:
Number of electrons = q_final / (-1.6 x 10^-19 coulombs)
By substituting the values into the equation, you can find the number of electrons that need to be removed from one sphere and added to the other sphere to create an attractive force of 0.9 Newton between them.