Charge q1 = 7.00 μC is at the origin, and charge q2 = -5.00 μC is on the x-axis, 0.300 m from the origin as shown below. (a) Find the magnitude and direction of the electric field at point P, which has coordinates (0, 0.400) m. (b) Find the force on a charge of 2.00 × 10-8 C placed at P.
To find the magnitude and direction of the electric field at point P, we can use Coulomb's law. Coulomb's law states that the electric field created by a point charge is given by:
E = k * |q| / r^2
where E is the electric field, k is the electrostatic constant (k = 9.0 x 10^9 N m^2/C^2), |q| is the magnitude of the point charge, and r is the distance from the point charge to the point where the electric field is being measured.
(a) Let's calculate the electric field at point P:
Distance from q1 to P:
r1 = √((0 - 0)^2 + (0.400 - 0)^2) = 0.400 m
Electric field due to q1 at P:
E1 = k * |q1| / r1^2 = (9.0 x 10^9 N m^2/C^2) * (7.00 x 10^-6 C) / (0.400 m)^2
Distance from q2 to P:
r2 = √((0.300 - 0)^2 + (0.400 - 0)^2) = 0.500 m
Electric field due to q2 at P:
E2 = k * |q2| / r2^2 = (9.0 x 10^9 N m^2/C^2) * (5.00 x 10^-6 C) / (0.500 m)^2
The total electric field at P is the vector sum of the electric fields due to q1 and q2:
E_total = E1 + E2
To find the magnitude and direction of E_total, we can calculate its x and y components:
Ex = E1 * cos(θ1) + E2 * cos(θ2)
Ey = E1 * sin(θ1) + E2 * sin(θ2)
where θ1 and θ2 are the angles between the x-axis and the line connecting q1 and P, and q2 and P, respectively. Since θ1 = 0 degrees (due to symmetry) and θ2 = atan(0.400 / 0.300), we can substitute the values and calculate.
Magnitude of E_total:
|E_total| = √(Ex^2 + Ey^2)
Direction of E_total:
θ_total = atan(Ey / Ex)
(b) To find the force on a charge of 2.00 x 10^-8 C placed at P, we can use the formula:
F = q * E
where F is the force, q is the charge, and E is the electric field.
F = (2.00 x 10^-8 C) * E_total
This will give us the force on the charge at point P.