The point charges in Figure 17.41 are located at the corners of an equilateral triangle 25.0 cm on a side, where qb = +12.0 µC and qc = -4.50 µC. (Assume that the +x-axis is directed to the right.)

(a) Find the electric field at the location of qa.
Magnitude
? N/C
Direction
? ° (counterclockwise from the +x-axis)

(b) What is the force on qa, given that qa = +1.50 nC?
Magnitude
? N
Direction
? ° (counterclockwise from the +x-axis)

To find the electric field at the location of qa, we can use the superposition principle and calculate the electric field contributions from each individual charge.

(a)
The magnitude of the electric field due to a point charge can be found using Coulomb's Law:

Electric field = k * (charge) / (distance^2)

Given:
qb = +12.0 µC
qc = -4.50 µC
Side length of equilateral triangle = 25.0 cm

First, we need to find the distances between the charges and the location of qa.

The distance between a corner charge and the center of the equilateral triangle can be found using trigonometry.
The distance is given by:
distance = (side length) * (√3 / 2)

distance = (25.0 cm) * (√3 / 2)
distance = 21.65 cm

The distance between qa and qb or qc is equal to the distance between the corner charge and the center of the equilateral triangle.

Using Coulomb's Law, we can calculate the electric field due to qb and qc individually.

Electric field due to qb:
Magnitude of charge = +12.0 µC = 12.0 * 10^-6 C
Distance = 21.65 cm = 0.2165 m
Electric field due to qb = (9 * 10^9 Nm^2/C^2) * (12.0 * 10^-6 C) / (0.2165 m)^2

Similarly, we can calculate the electric field due to qc:

Magnitude of charge = -4.50 µC = -4.50 * 10^-6 C
Electric field due to qc = (9 * 10^9 Nm^2/C^2) * (-4.50 * 10^-6 C) / (0.2165 m)^2

To find the net electric field at the location of qa, we need to add the individual electric fields due to qb and qc considering their signs.

Net Electric Field = Electric field due to qb + Electric field due to qc

Once you calculate the magnitudes of these electric fields, you can find the direction using the angles in the equilateral triangle.

(b)
To find the force on qa, we can use the equation:

Force = (charge) * (electric field)

Given:
qa = +1.50 nC = 1.50 * 10^-9 C

Using the net electric field calculated in part (a), we can find the force on qa by multiplying the charge qa with the electric field.

Force = (1.50 * 10^-9 C) * (Net Electric Field)

The magnitude and direction of the force can be found using the given net electric field and the direction of the electric field at the location of qa.

To find the electric field at the location of charge qa, you can use the principle of superposition. This principle states that the total electric field at any point is the vector sum of the individual electric fields created by each charge.

(a) To calculate the electric field at the location of qa:

1. Divide the equilateral triangle into two right triangles.
2. For each right triangle, calculate the electric field at the center using Coulomb's law.

To calculate the electric field due to qb:
- The distance between qb and the center is the length of the side of the equilateral triangle divided by 2 (√(3)/2 * 25 cm).
- Use the formula E = k * |qb| / r^2, where k is the electrostatic constant (9 x 10^9 Nm^2/C^2), |qb| is the magnitude of the charge, and r is the distance between qb and the center.

To calculate the electric field due to qc:
- The distance between qc and the center can be found using the Pythagorean theorem.
- Use the formula E = k * |qc| / r^2 to calculate the electric field at the center due to qc.

3. Add the electric fields due to qb and qc as vectors to find the total electric field at the center of the triangle.

(b) To calculate the force on qa:

- The force acting on a charged particle in an electric field is given by the equation F = q * E, where F is the force, q is the charge of the particle, and E is the electric field.
- Substitute the values of qa and the electric field calculated in part (a) into this equation.
- The magnitude of the force is given by |F| = |q| * |E|.
- The direction of the force can be determined by the angle between the force vector and the positive x-axis. You can find this angle using trigonometry.

Perform these calculations to find the electric field and force at the location of charge qa.

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