Eight charges (50μC each) are arranged on the corners of a cube side length 1 cm. What is the magnitude of the force on each charge in Newtons?
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To find the magnitude of the force on each charge, we can use Coulomb's Law, which states that the force between two charges is given by the equation:
F = k * (q1 * q2) / r^2,
where F is the force between the charges, k is the Coulomb's constant (k = 9 × 10^9 N·m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
In this case, we have 8 charges, so we need to calculate the force on each charge due to the other 7 charges.
1. Let's label the charges as A, B, C, D, E, F, G, and H, with A being the charge at the bottom left corner, and the charges labeled clockwise.
2. For each charge, we can calculate the magnitude of the force by summing up the forces due to the other 7 charges.
To calculate the net force on charge A:
Force on A = Force AB + Force AC + Force AD + Force AE + Force AF + Force AG + Force AH
1. Force AB is the force between charge A and charge B.
The distance between A and B is the length of the side of the cube = 1 cm = 0.01 m.
Plugging in the values, we get:
Force AB = (9 × 10^9 N·m^2/C^2) * ((50 × 10^-6 C) * (50 × 10^-6 C)) / (0.01 m)^2
2. Repeat this process for the forces between A and C, A and D, A and E, A and F, A and G, and A and H.
3. The net force on charge A will be the sum of all these forces. Repeat this process for all the other charges (B, C, D, E, F, G, and H).
Since these calculations involve a lot of steps, it might be best to use a calculator or a computer program to simplify the process.
To find the magnitude of the force on each charge, we can use Coulomb's law. Coulomb's law states that the force between two charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Given that each charge has a magnitude of 50μC, we can convert this to Coulombs by using the conversion factor 1μC = 10^(-6) C. Therefore, each charge has a magnitude of 50 x 10^(-6) C.
The distance between any two charges on the corners of a cube can be found using Pythagoras' theorem. For a cube with side length 1 cm, the distance between charges on the adjacent corners can be calculated as the diagonal of the cube's face, which is √(1^2 + 1^2) = √2 cm.
Now, we can plug in these values to Coulomb's law:
F = k * (q1 * q2) / r^2
where:
F is the magnitude of the force,
k is the electrostatic constant (k = 8.99 x 10^9 N m^2/C^2),
q1 and q2 are the charges,
and r is the distance between the charges.
In this case, we are calculating the force on each charge induced by the other seven charges. Since all charges are the same, we can calculate the force on one charge and that will be the same for all charges. Let's calculate the force using the formula above:
F = (8.99 x 10^9 N m^2/C^2 ) * [(50 x 10^(-6) C) * (50 x 10^(-6) C)] / (√2 cm)^2
Simplifying the equation, we get:
F = (8.99 x 10^9 N m^2/C^2 ) * (2.5 x 10^(-3) C^2) / 2 cm^2
Finally, we can convert cm to meters (1 cm = 0.01 m) and calculate the force:
F = (8.99 x 10^9 N m^2/C^2 ) * (2.5 x 10^(-3) C^2) / (0.01 m)^2
F ≈ 8.99 x 10^9 * 2.5 x 10^(-3) / 0.01^2
F ≈ 2.247 x 10^(-3) N
Therefore, the magnitude of the force on each charge is approximately 2.247 x 10^(-3) Newtons.