Four point charges are at the corners of a square of side a as shown in the figure below. Determine the magnitude and direction of the resultant electric force on q, with ke, q, and a left in symbolic form. (B = 4q and C = 6.5q. Let the +x-axis be pointing to the right.)
b|---a----|q
| |
a a
| |
c|_ _ _ a _ _|b
Easy. the adjacent charges both b have forces at 90 degrees, so they add equally diagonally F=2k(4q^2)/a^2 * sin45
then add to that the diagonal force
Fd= k6.5q^2/(asqrt2)^2
now add those two forces. Your teacher is top easy.
To determine the magnitude and direction of the resultant electric force on q, we need to calculate the individual electric forces between q and each of the other charges and then add them vectorially.
The electric force between two charges is given by Coulomb's Law:
Fe = ke * (q1 * q2) / r^2
Where:
- ke is the Coulomb's constant
- q1 and q2 are the magnitudes of the charges
- r is the distance between the charges
Let's calculate the electric force between q and each of the other charges:
1. Between q and charge B:
The distance between q and B is a. Therefore, the electric force between them is:
Fe1 = ke * (q * 4q) / (a^2)
2. Between q and charge C:
The distance between q and C is √(2a)^2 = 2a. Therefore, the electric force between them is:
Fe2 = ke * (q * 6.5q) / ((2a)^2)
Now, let's calculate the components of these forces along the x-axis:
1. Fe1x:
The force Fe1 makes an angle of 45 degrees with the x-axis (since it is a square). Therefore, Fe1x = Fe1 * cos(45)
2. Fe2x:
The force Fe2 makes an angle of 135 degrees with the x-axis (since it is a square). Therefore, Fe2x = Fe2 * cos(135)
Next, we add the components of the forces along the x-axis to determine the resultant force:
Resultant force on q along the x-axis = Fe1x + Fe2x
Finally, to determine the magnitude and direction of the resultant force, we calculate the magnitude using Pythagorean theorem, and the direction using the inverse tangent function:
Magnitude of Resultant force = sqrt((Fe1x + Fe2x)^2)
Direction of Resultant force = atan((Fe1x + Fe2x) / 0) (as the +x-axis is pointing to the right)
Note: Please note that the given values of ke, q, and a are not mentioned in the question, so you will need to substitute those values and perform the calculations to obtain the final numerical result.
To determine the magnitude and direction of the resultant electric force on q, we need to calculate the electric force due to each individual charge and then add them up vectorially.
Step 1: Calculate the electric force due to charge B on q.
The formula for the electric force between two point charges is given by Coulomb's Law:
F = (k * |q1 * q2|) / r^2
Here, q1 is the charge on q, q2 is the charge on B, r is the distance between them, and k is the Coulomb's constant.
The distance between q and B is a, and the electric force between them is attractive since they have opposite charges. Therefore, the direction will be towards charge B, which is along the -x-axis.
So, the electric force due to B on q is:
F_Bq = (k * |q * 4q|) / a^2
Step 2: Calculate the electric force due to charge C on q.
Similar to the previous step, we can calculate the electric force between q and C using Coulomb's Law.
The distance between q and C is √(a^2 + a^2) = √2a, and the electric force between them is attractive since they have opposite charges. Therefore, the direction will be towards charge C, which is along the +y-axis.
So, the electric force due to C on q is:
F_Cq = (k * |q * 6.5q|) / (2a)^2
Step 3: Determine the resultant force between B and C.
Since the x and y components of the forces due to B and C act along different axes, we can calculate them separately.
The x-component of F_Bq is F_Bq_x = -F_Bq * cos(90°) = -F_Bq * 0 = 0.
The y-component of F_Cq is F_Cq_y = F_Cq * sin(90°) = F_Cq * 1 = F_Cq.
Step 4: Calculate the magnitude and direction of the resultant force.
The magnitude of the resultant force is given by the Pythagorean theorem:
|F_res| = √(F_Bq_x^2 + F_Cq_y^2)
Since F_Bq_x = 0, the magnitude simplifies to:
|F_res| = |F_Cq_y|
The direction of the resultant force is given by the inverse tangent:
θ = tan^(-1)(F_Cq_y / F_Bq_x)
In this case, since F_Bq_x = 0, the direction simplifies to:
θ = tan^(-1)(F_Cq / F_Bq_x)
Therefore, to determine the magnitude and direction of the resultant electric force on q, you would need to plug in the appropriate values for ke, q, and a into the formulas above and calculate the result.