A test charge of +2 μC is placed halfway between a charge of +6 μC and another of +4 μC separated by 10 cm. What is the magnitude of the force on the test charge?
To calculate the magnitude of the force on the test charge, we can use Coulomb's Law. Coulomb's Law states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Let's break it down step-by-step:
Step 1: Determine the distances between the test charge and the two other charges.
In this case, the test charge is placed halfway between the +6 μC charge and the +4 μC charge, so the distance between the test charge and each of these charges is 5 cm (half of the total separation distance of 10 cm).
Step 2: Calculate the magnitude of the force between the test charge and each of the other charges individually.
Using Coulomb's Law, we can calculate the force F1 between the test charge and the +6 μC charge and the force F2 between the test charge and the +4 μC charge.
The equation for Coulomb's Law is:
F = k * (q1 * q2) / r^2
where F is the magnitude of the force, k is the electrostatic constant (9.0 x 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
For the force between the test charge and the +6 μC charge:
F1 = (9.0 x 10^9 N m^2/C^2) * ((2 μC) * (6 μC)) / (0.05 m)^2
For the force between the test charge and the +4 μC charge:
F2 = (9.0 x 10^9 N m^2/C^2) * ((2 μC) * (4 μC)) / (0.05 m)^2
Step 3: Calculate the total force on the test charge.
To find the total force on the test charge, we add the individual forces together:
Total force = F1 + F2
Step 4: Calculate the magnitude of the total force.
To obtain the magnitude of the total force, we take the absolute value of the total force:
Magnitude of force = |Total force|
Simplifying and calculating the values, we get:
F1 = (9.0 x 10^9 N m^2/C^2) * (12 x 10^-6 C^2) / (0.05 m)^2
= (9.0 x 10^9 N m^2/C^2) * (12 x 10^-6 C^2) / 0.0025 m^2
= (9.0 x 10^9 N m^2/C^2) * 4.8 x 10^-3 C^2 / m^2
= 43.2 N
F2 = (9.0 x 10^9 N m^2/C^2) * (8 x 10^-6 C^2) / (0.05 m)^2
= (9.0 x 10^9 N m^2/C^2) * (8 x 10^-6 C^2) / 0.0025 m^2
= (9.0 x 10^9 N m^2/C^2) * 3.2 x 10^-3 C^2 / m^2
= 28.8 N
Total force = F1 + F2
= 43.2 N + 28.8 N
= 72 N
Magnitude of force = |Total force|
= |72 N|
= 72 N
Therefore, the magnitude of the force on the test charge is 72 N.
To calculate the magnitude of the force on the test charge, we can use Coulomb's law. Coulomb's law states that the magnitude of 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.
The formula for Coulomb's law is:
F = k * (|q1| * |q2|) / r^2
where:
F is the magnitude of the force
k is the electrostatic constant (k = 9 * 10^9 N m²/C²)
|q1| and |q2| are the magnitudes of the charges
r is the distance between the charges
In this case, we have a test charge of +2 μC, a charge of +6 μC, and a charge of +4 μC. The distance between the test charge and each of the other charges is 10 cm.
Let's calculate the magnitude of the force on the test charge using Coulomb's law:
Step 1: Convert the charges from microcoulombs to coulombs.
+2 μC = 2 * 10^-6 C
+6 μC = 6 * 10^-6 C
+4 μC = 4 * 10^-6 C
Step 2: Convert the distance from centimeters to meters.
10 cm = 10 * 10^-2 m
Step 3: Plug the values into the Coulomb's law formula.
F = (9 * 10^9 N m²/C²) * ((2 * 10^-6 C) * (6 * 10^-6 C)) / (10 * 10^-2 m)²
Simplifying this expression gives us:
F = (9 * 10^9 N m²/C²) * (12 * 10^-12 C²) / (100 * 10^-4 m²)
F = (108 * 10^-3 N m²) / (100 * 10^-4 m²)
F = 108 * (10^-3 N m²) * (10^4 m²) / 100
F = 1080 N
Therefore, the magnitude of the force on the test charge is 1080 N.