A +48.00 nC point charge is placed on the x-axis at x= 4.000 m. and a-24.00-nc point charge is placed on the y-axis at y=-6.000 m. What is the direction of the electric field at the origin?

Question

in simple steps

Answer 1

To find the direction of the electric field at the origin, we need to calculate the electric field due to each charge individually and then add them to get the total electric field at the origin.

Step 1: Calculate the electric field due to the +48.00 nC charge:
- The electric field due to a point charge is given by the equation E = k * Q / r^2, where E is the electric field, k is the Coulomb constant (9 * 10^9 N m^2/C^2), Q is the charge, and r is the distance from the charge.
- Given Q = 48.00 nC = 48.00 * 10^-9 C and r = 4.000 m, we can plug in these values into the equation to find the electric field due to the +48.00 nC charge.
- E = (9 * 10^9 N m^2/C^2) * (48.00 * 10^-9 C) / (4.000 m)^2
- E = 10.8 N/C (approximately)

Step 2: Calculate the electric field due to the -24.00 nC charge:
- Using the same equation, E = k * Q / r^2, we can calculate the electric field due to the -24.00 nC charge.
- Given Q = -24.00 nC = -24.00 * 10^-9 C and r = 6.000 m (distance from the charge to the origin), we can plug in these values into the equation.
- E = (9 * 10^9 N m^2/C^2) * (-24.00 * 10^-9 C) / (6.000 m)^2
- E = -12.0 N/C (approximately)

Step 3: Add the electric fields due to each charge:
- Since the electric field is a vector quantity, we need to add the magnitudes and consider the direction.
- The electric field due to the +48.00 nC charge is to the right (positive x direction), and the electric field due to the -24.00 nC charge is to the downward (negative y direction).
- Therefore, the total electric field at the origin is the vector sum of these two electric fields, which can be visualized as the diagonal of the resultant electric field vector.
- Since the magnitudes of the electric fields are equal (10.8 N/C and 12.0 N/C), and they form a right-angled triangle, we can use the Pythagorean theorem to find the magnitude of the total electric field.
- Magnitude of the total electric field = sqrt((10.8 N/C)^2 + (12.0 N/C)^2) = 16.7 N/C (approximately).

Step 4: Determine the direction of the electric field:
- To find the direction of the electric field, we can use trigonometry.
- The angle between the resultant electric field vector and the positive x direction can be found using the inverse tangent function.
- Angle = atan((12.0 N/C) / (10.8 N/C)) = 48.2 degrees (approximately).
- Therefore, the direction of the electric field at the origin is approximately 48.2 degrees below the positive x direction, or in the fourth quadrant.