"Charges of 2.00uc and -3.00uc are located in a coordinate plane at(2.00cm, 5.00 cm) and (3.00cm, 2.00cm) respectively. calculate the magnitude and direction of the electric field created by this pair at the origin. express your answer in both polar and rectangular form"
To solve this problem, we can use Coulomb's Law to calculate the electric field created by each charge at the origin.
Coulomb's Law states that the magnitude of the electric field (E) created by a point charge is given by the equation:
E = k * (|q| / r^2)
where:
- E is the electric field magnitude
- k is Coulomb's constant (9.0 x 10^9 N m^2 / C^2)
- |q| is the magnitude of the charge
- r is the distance between the charge and the point where the electric field is being measured
Let's calculate the electric field for each charge separately and then determine the net electric field at the origin.
1. Electric Field due to the first charge:
Magnitude of the first charge (|q1|) = 2.00 uc = 2.00 x 10^-6 C
Distance between the first charge and the origin: r1 = sqrt((2.00 cm)^2 + (5.00 cm)^2) = 5.39 cm = 0.0539 m
Using Coulomb's Law for the first charge:
E1 = (9.0 x 10^9 N m^2 / C^2) * (|q1| / r1^2)
= (9.0 x 10^9 N m^2 / C^2) * (2.00 x 10^-6 C / (0.0539 m)^2)
= 6.11 x 10^5 N/C
2. Electric Field due to the second charge:
Magnitude of the second charge (|q2|) = -3.00 uc = -3.00 x 10^-6 C
Distance between the second charge and the origin: r2 = sqrt((3.00 cm)^2 + (2.00 cm)^2) = 3.61 cm = 0.0361 m
Using Coulomb's Law for the second charge:
E2 = (9.0 x 10^9 N m^2 / C^2) * (|q2| / r2^2)
= (9.0 x 10^9 N m^2 / C^2) * (3.00 x 10^-6 C / (0.0361 m)^2)
= 2.46 x 10^6 N/C
3. Net Electric Field at the origin:
Since the electric field is a vector quantity, we need to consider the direction as well. Adding the electric fields due to each charge separately, we get:
Net electric field (E_net) = E1 + E2
To express the answer in rectangular form, we need to combine the x and y components of the electric field at the origin:
E_x = E_net * cos(theta)
E_y = E_net * sin(theta)
To express the answer in polar form, we can determine the magnitude (E) and the angle (theta) using:
E = sqrt(E_x^2 + E_y^2)
theta = arctan(E_y / E_x)
Now, let's calculate the net electric field and its direction:
Net electric field (E_net) = E1 + E2
= 6.11 x 10^5 N/C + 2.46 x 10^6 N/C
= 3.07 x 10^6 N/C
To calculate the x and y components of the electric field at the origin:
E_x = E_net * cos(theta)
E_x = 3.07 x 10^6 N/C * cos(theta)
E_y = E_net * sin(theta)
E_y = 3.07 x 10^6 N/C * sin(theta)
To express the answer in polar form:
E = sqrt(E_x^2 + E_y^2)
E = sqrt((3.07 x 10^6 N/C * cos(theta))^2 + (3.07 x 10^6 N/C * sin(theta))^2)
theta = arctan(E_y / E_x)
theta = arctan((3.07 x 10^6 N/C * sin(theta)) / (3.07 x 10^6 N/C * cos(theta)))
You can substitute the appropriate values for theta into these equations to calculate the polar and rectangular forms of the net electric field at the origin.