two identical copper spheres are separated by 1m in vacuum. how many electrons would have to be removed from one sphere and added to other sphere so that they attract each other with force of 0.9N
To determine the number of electrons that would need to be removed from one copper sphere and added to the other, we need to use Coulomb's law, which states that the force of attraction or repulsion between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Let's break down the steps to find the answer:
Step 1: Gather the necessary information.
We are given:
- The force of attraction between the two spheres, which is 0.9 N.
- The distance between the spheres, which is 1 m.
- The spheres are made of copper.
Step 2: Finding the charge of each sphere.
The charge on an electron is approximately -1.6 x 10^-19 C (Coulombs). However, copper is not a perfect conductor, so not all of its electrons are free to move. In a copper sphere, we can assume that each copper atom contributes one free electron.
Step 3: Calculating the charge on each sphere.
Since the spheres are identical, let's assume that one sphere loses x electrons (represented by -x) while the other sphere gains x electrons (represented by +x).
The net charge on each sphere can be calculated using the formula:
Charge = Number of electrons x Charge per electron.
For the sphere losing electrons:
Charge (Sphere 1) = -x (number of electrons lost) x (-1.6 x 10^-19 C) (charge per electron)
For the sphere gaining electrons:
Charge (Sphere 2) = x (number of electrons gained) x (-1.6 x 10^-19 C) (charge per electron)
Step 4: Calculating the force of attraction using Coulomb's law.
Coulomb's law equation is: F = k * (|q1 * q2| / r^2),
Where:
F is the force of attraction,
k is Coulomb's constant (approximately 9 x 10^9 Nm^2/C^2),
|q1 * q2| is the absolute value of the product of the charges, and
r is the distance between the centers of the two spheres.
We can rearrange this equation to solve for the charge (|q1 * q2|) required for the force of attraction to be 0.9 N.
|q1 * q2| = (F * r^2) / k
Substituting the given values in:
|q1 * q2| = (0.9 N * (1 m)^2) / (9 x 10^9 Nm^2/C^2)
Calculating |q1 * q2| will give us the charge needed for the force of attraction.
Step 5: Solving for the number of electrons.
Now, we can equate the calculated charge value (|q1 * q2|) to the charge difference between Sphere 1 and Sphere 2:
|x * (-1.6 x 10^-19 C)| = (0.9 N * (1 m)^2) / (9 x 10^9 Nm^2/C^2)
Solving for x will give us the number of electrons that need to be transferred between the spheres.
Step 6: Calculate the number of electrons.
To find x, we can rearrange the equation as follows:
|x| = [(0.9 N * (1 m)^2) / (9 x 10^9 Nm^2/C^2)] / (1.6 x 10^-19 C)
Simplify the equation by canceling units and solving for x.
|x| = 0.1 x 10^10
Take the absolute value of x because we are working with magnitudes of charges.
Therefore, x ≈ 1 x 10^10 = 10,000,000,000 (ten billion) electrons need to be transferred.
Thus, to attract each other with a force of 0.9 N, 10 billion electrons would need to be removed from one sphere and added to the other.