Two small spheres spaced 20.0 centimeters apart have equal charge.
How many excess electrons must be present on each sphere if the magnitude of the force of repulsion between them is 4.57×10 −21 newtons?
To determine the number of excess electrons on each sphere, we need to use Coulomb's law, which relates the magnitude of the electrostatic force between two charged objects to the charges and distance between them.
Coulomb's law equation: F = k * (|q₁| * |q₂|) / r²
Where:
F is the magnitude of the electrostatic force
k is the electrostatic constant (9 × 10⁹ N m²/C²)
|q₁| and |q₂| are the magnitudes of the charges on the spheres
r is the distance between the spheres
In this case, the two spheres have the same charge magnitude (q₁ = q₂), and the force of repulsion (F) is given as 4.57 × 10⁻²¹ N. The distance (r) between the spheres is 20.0 cm, which can be converted to meters (0.20 m) for consistency.
Rearranging the equation, we can solve for the charge magnitude (|q₁|):
|q₁| = (F * r²) / (k)
Substituting the given values:
|q₁| = (4.57 × 10⁻²¹ N * (0.20 m)²) / (9 × 10⁹ N m²/C²)
Calculating the expression:
|q₁| ≈ 2.55 × 10⁻¹⁹ C
Since the two spheres have the same charge, each sphere would have an excess of 2.55 × 10⁻¹⁹ C of charge.
To determine the number of excess electrons, we can use the elementary charge (e) which is equal to 1.6 × 10⁻¹⁹ C. Dividing the charge magnitude by the elementary charge:
Number of excess electrons = |q₁| / e
Substituting the values:
Number of excess electrons ≈ (2.55 × 10⁻¹⁹ C) / (1.6 × 10⁻¹⁹ C)
Calculating the expression:
Number of excess electrons ≈ 1.59
Therefore, there are approximately 1.59 × 10¹ excess electrons on each sphere.
To find the number of excess electrons on each sphere, we can use Coulomb's Law, which states that the force of 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.
1. First, let's convert the distance between the spheres to meters:
20.0 centimeters = 20.0 / 100 = 0.20 meters
2. Coulomb's Law equation is:
F = k * (q1 * q2) / r^2
where F is the force of repulsion, k is the electrostatic constant, q1 and q2 are the charges on the spheres, and r is the distance between the spheres.
3. Rearrange the equation to solve for the product of the charges:
(q1 * q2) = (F * r^2) / k
We need the product of the charges because the spheres have equal charge.
4. Substitute the known values into the equation:
(q1 * q2) = (4.57 × 10^(-21) N * (0.20 m)^2) / (9 × 10^9 N m^2/C^2)
5. Calculate the product of the charges:
(q1 * q2) = (4.57 × 10^(-21) N * 0.04 m^2) / (9 × 10^9 N m^2/C^2)
(q1 * q2) = 2.028 × 10^(-22) C^2
6. Since the spheres have the same charge, we can assume q1 = q2 = q.
Therefore, q^2 = 2.028 × 10^(-22) C^2
7. Take the square root of both sides to solve for q:
q = √(2.028 × 10^(-22) C^2)
q ≈ 4.50 × 10^(-12) C
8. Each electron has a charge of -1.60 × 10^(-19) C.
9. Calculate the number of excess electrons on each sphere:
number of electrons = (charge of sphere) / (charge of one electron)
number of electrons = (4.50 × 10^(-12) C) / (-1.60 × 10^(-19) C)
number of electrons ≈ -2.81 × 10^7
Since the charge is negative, it means there is an excess of electrons on each sphere. Therefore, each sphere has approximately 2.81 × 10^7 excess electrons.
F=k QQ/r^2
solve for q, then dvide by the charge on one electron