Two identical positive charges exert a repulsive force of 7.0 10-9 N when separated by a distance of 3.2 10-10 m. Calculate the charge of each.
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To calculate the charge of each positive charge, we can use Coulomb's Law, which states that the force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
Let's break down the steps to solve this problem:
Step 1: Identify the given values:
- Repulsive force between the charges (F) = 7.0 x 10^-9 N
- Distance between the charges (r) = 3.2 x 10^-10 m
Step 2: Identify the formula related to the problem:
Coulomb's Law states that F = k * (q1 * q2) / r^2
where:
- F is the force between the charges
- k is the electrostatic constant (9 x 10^9 Nm^2/C^2)
- q1 and q2 are the magnitudes of the charges
- r is the distance between the charges
Step 3: Rearrange the formula to solve for charge (q):
We can rearrange Coulomb's Law to solve for q1 or q2:
q1 = (F * r^2) / (k * q2)
q2 = (F * r^2) / (k * q1)
Step 4: Plug in the values and calculate:
Using our rearranged formula, let's calculate the value of q1:
q1 = (7.0 x 10^-9 N * (3.2 x 10^-10 m)^2) / (9 x 10^9 Nm^2/C^2 * q2)
q1 = 7.414 x 10^-20 C / q2
Similarly, we can calculate the value of q2 using the same formula:
q2 = (7.0 x 10^-9 N * (3.2 x 10^-10 m)^2) / (9 x 10^9 Nm^2/C^2 * q1)
q2 = 7.414 x 10^-20 C / q1
Step 5: Calculate the values of q1 and q2:
We need both equations to find the values of q1 and q2. We can solve this system of equations by substituting one equation into the other.
Let's solve the equations numerically:
q1 = 7.414 x 10^-20 C / q2
q2 = 7.414 x 10^-20 C / (7.414 x 10^-20 C / q2)
Simplifying gives:
q1 = 1 C
q2 = 1 C
Therefore, each charge is equal to 1 Coulomb (C).
To calculate the charge of each particle, we can use Coulomb's Law, which states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Let's assume that the charge of each particle is q. Therefore, the force between the two particles can be written as:
F = k * (q * q) / r^2
Where:
F is the force between the charges
k is the electrostatic constant (9 * 10^9 N m^2 / C^2)
q is the charge of each particle
r is the distance between the particles
We can rearrange the formula to solve for q:
q * q = (F * r^2) / k
Substituting in the given values:
F = 7.0 * 10^-9 N
r = 3.2 * 10^-10 m
k = 9 * 10^9 N m^2 / C^2
q * q = (7.0 * 10^-9 N * (3.2 * 10^-10 m)^2) / (9 * 10^9 N m^2 / C^2)
Now, let's calculate the charge of each particle using this formula:
q * q = (7.0 * 10^-9 N * (3.2 * 10^-10 m)^2) / (9 * 10^9 N m^2 / C^2)
q * q = 7.168 * 10^-27 C^2
Taking the square root of both sides, we get:
q = sqrt(7.168 * 10^-27 C^2)
Calculating this, we find:
q ≈ 2.68 * 10^-14 C
Therefore, the charge of each particle is approximately 2.68 * 10^-14 Coulombs.
F=k•q₁•q₂/r²= k•q²/r²
q=sqrt{ F•r²/k}=sqrt{7•10⁻⁹•(3.2•10⁻¹⁰)²/9•10⁹}=2.82•10⁻¹⁸C