Three charged particles are placed at the corners of an equilateral triangle of side 1.20 m (see figure). The charges are Q1 = +3.8 µC, Q2 = -9.0 µC, and Q3 = -6.1 µC. Calculate the magnitude and direction of the net force on each due to the other two. (Assume the +x axis points to the right, that is, from Q2 toward Q3.)
To calculate the magnitude and direction of the net force on each charged particle, we can use Coulomb's Law which states that the force between two charged particles is proportional to the product of their charges and inversely proportional to the square of the distance between them. The formula for Coulomb's Law is:
F = k * (|q1| * |q2|) / r^2
Where:
F is the force between the charged particles,
k is the electrostatic constant, approximately 9.0 x 10^9 Nm^2/C^2,
q1 and q2 are the magnitudes of the charges,
r is the distance between the charges.
Let's calculate the net forces on each particle step by step:
1. Net force on Q1:
Q1 experiences forces from both Q2 and Q3. Since Q2 and Q3 have opposite charges, the forces on Q1 will be in opposite directions.
The magnitude of the force between Q1 and Q2 is:
F1,2 = k * (|Q1| * |Q2|) / r^2, where r is the length of the side of the equilateral triangle.
The direction of F1,2 is from Q2 towards Q1.
The magnitude of the force between Q1 and Q3 is:
F1,3 = k * (|Q1| * |Q3|) / r^2, where r is the length of the side of the equilateral triangle.
The direction of F1,3 is from Q3 towards Q1.
To calculate the net force on Q1, we need to find the vector sum of F1,2 and F1,3.
Net Force on Q1 = F1,2 + F1,3
2. Net force on Q2:
Q2 also experiences forces from both Q1 and Q3. The directions of the forces depend on the charges.
The magnitude of the force between Q2 and Q1 is:
F2,1 = k * (|Q2| * |Q1|) / r^2, where r is the length of the side of the equilateral triangle.
The direction of F2,1 is from Q2 towards Q1.
The magnitude of the force between Q2 and Q3 is:
F2,3 = k * (|Q2| * |Q3|) / r^2, where r is the length of the side of the equilateral triangle.
The direction of F2,3 is from Q2 towards Q3.
To calculate the net force on Q2, we need to find the vector sum of F2,1 and F2,3.
Net Force on Q2 = F2,1 + F2,3
3. Net force on Q3:
Q3 experiences forces from both Q1 and Q2.
The magnitude of the force between Q3 and Q1 is:
F3,1 = k * (|Q3| * |Q1|) / r^2, where r is the length of the side of the equilateral triangle.
The direction of F3,1 is from Q3 towards Q1.
The magnitude of the force between Q3 and Q2 is:
F3,2 = k * (|Q3| * |Q2|) / r^2, where r is the length of the side of the equilateral triangle.
The direction of F3,2 is from Q3 towards Q2.
To calculate the net force on Q3, we need to find the vector sum of F3,1 and F3,2.
Net Force on Q3 = F3,1 + F3,2
By calculating the above expressions, we can find the magnitude and direction of the net force on each charged particle due to the other two.
To calculate the magnitude and direction of the net force on each charged particle due to the other two, we can use Coulomb's Law. Coulomb's Law 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 calculate the net force on each charged particle step-by-step:
Step 1: Calculate the distance between the charged particles.
The side length of the equilateral triangle is given as 1.20 m. Since all sides of an equilateral triangle are equal, the distance between any two charged particles will be the same.
Step 2: Calculate the force between Q1 and Q2.
The force between Q1 and Q2 is given by Coulomb's Law:
F1,2 = (k * |Q1 * Q2|) / r^2
Where k is the electrostatic constant (k = 8.99 x 10^9 N m^2/C^2), |Q1| and |Q2| are the absolute values of the charges, and r is the distance between them.
Substituting the given values:
|Q1| = 3.8 x 10^-6 C
|Q2| = 9.0 x 10^-6 C
r = 1.20 m
F1,2 = (8.99 x 10^9 N m^2/C^2 * |3.8 x 10^-6 C| * |9.0 x 10^-6 C|) / (1.20 m)^2
Calculating F1,2 will give you the magnitude of the force between Q1 and Q2.
Step 3: Calculate the force between Q1 and Q3.
Follow the same steps as above, substituting the values for Q1, Q3, and r.
Step 4: Calculate the force between Q2 and Q3.
Follow the same steps as above, substituting the values for Q2, Q3, and r.
Step 5: Calculate the net force on Q1.
Since Q1 is charged positively (+3.8 µC), the direction of the net force will be towards the other two negatively charged particles.
To find the net force, add the vectors of the individual forces acting on Q1. Consider the directions and magnitudes of the forces to determine whether the net force is in the positive x-direction or negative x-direction.
Step 6: Calculate the net force on Q2.
Since Q2 is negatively charged (-9.0 µC), the direction of the net force will be in the positive x-direction (from Q2 toward Q3).
Step 7: Calculate the net force on Q3.
Since Q3 is also negatively charged (-6.1 µC), the direction of the net force will be in the negative x-direction (from Q3 toward Q2).
Follow the same steps as above to calculate the net force on Q3.
Use coulombs law for each force. Then, break up each of the forces into x,y components. You will have then six components.
Now, look at each corner. Add the two x, and the two y, calculate the magnitude and direction from those two components.
Repeat for each corner.
Remember, think about direction. two like charges repel, two unlike attract.