Two point charges are placed on the x axis. (Figure 1) The first charge, q1 = 8.00nC , is placed a distance 16.0m from the origin along the positive x axis; the second charge, q2 = 6.00nC , is placed a distance 9.00m from the origin along the negative x axis.
Calculate the electric field at point A, located at coordinates (0 m, 12.0m ).
Give the x and y components of the electric field as an ordered pair. Express your answer in newtons per coulomb to three significant figures.
(2.20 N/C, -2.20 N/C)
To calculate the electric field at point A, we can use the principle of superposition. The electric field at point A due to q1 and q2 will be the sum of the individual electric fields generated by each charge.
The formula to calculate the electric field generated by a point charge is given by:
E = k * (q / r^2)
Where E is the electric field, k is the Coulomb's constant (k = 8.99 x 10^9 Nm^2/C^2), q is the charge, and r is the distance from the charge to the point where we want to calculate the electric field.
First, let's calculate the electric field generated by q1 at point A:
Distance from q1 to A, r1 = 12.0 m (distance along the y-axis)
Charge q1 = 8.00 nC
E1 = k * (q1 / r1^2)
Substituting the values:
E1 = (8.99 x 10^9 Nm^2/C^2) * (8.00 x 10^-9 C) / (12.0 m)^2
Calculating that gives us:
E1 ≈ 2.497 x 10^3 N/C
Next, let's calculate the electric field generated by q2 at point A:
Distance from q2 to A, r2 = 12.0 m (distance along the y-axis)
Charge q2 = -6.00 nC (- sign indicates that the charge is negative)
Electric field generated by q2, E2 = k * (q2 / r2^2)
Substituting the values:
E2 = (8.99 x 10^9 Nm^2/C^2) * (-6.00 x 10^-9 C) / (12.0 m)^2
Calculating that gives us:
E2 ≈ -1.122 x 10^3 N/C
Now, since the electric field is a vector quantity, we need to consider the direction as well. As we can see, the electric field due to q1 is positive, while the electric field due to q2 is negative.
The total electric field at point A will be the sum of the electric fields generated by q1 and q2:
E_total = E1 + E2
Substituting the values:
E_total = 2.497 x 10^3 N/C + (-1.122 x 10^3 N/C)
Calculating that gives us:
E_total ≈ 1.375 x 10^3 N/C
So, the magnitude of the electric field at point A is approximately 1.375 x 10^3 N/C. However, we also need to determine the direction of the electric field at point A.
Since the electric field due to q1 is non-zero only along the y-axis, it will contribute only to the y-component of the electric field at point A.
Therefore, the x-component of the electric field at point A is zero.
In summary, the x-component of the electric field at point A is 0 N/C, and the y-component is approximately 1.375 x 10^3 N/C.
To calculate the electric field at point A, we can use Coulomb's law and superposition principle.
Step 1: Calculate the electric field due to charge q1 at point A.
The formula for the electric field due to a charge is given by:
E = k * q / r^2
where E is the electric field, k is the Coulomb's constant (8.99 x 10^9 N*m^2/C^2), q is the charge, and r is the distance between the charge and the point where the electric field is being calculated.
In this case, we are calculating the electric field at point A due to charge q1. The distance between q1 and point A is the y component of point A's coordinates, which is 12.0 m. The charge q1 is positive, so we can directly use its value.
Using the above formula, we get:
E1 = (8.99 x 10^9 N*m^2/C^2) * (8.00 x 10^-9 C) / (12.0 m)^2
Step 2: Calculate the electric field due to charge q2 at point A.
Similarly, we can calculate the electric field at point A due to charge q2. The distance between q2 and point A is the sum of the x component of q2's coordinates (9.0 m) and the x component of point A's coordinates (0 m). The charge q2 is negative, so we need to take its magnitude.
Using the above formula, we get:
E2 = (8.99 x 10^9 N*m^2/C^2) * (6.00 x 10^-9 C) / (9.0 + 0 m)^2
Step 3: Calculate the total electric field at point A.
To find the total electric field at point A, we need to add the x and y components of the electric fields due to q1 and q2 separately. The x component of the electric field due to q2 is in the negative direction.
Therefore, the x component of the electric field at point A is:
Ex = E1 - E2
The y component of the electric field at point A is the sum of the y components of the electric fields due to q1 and q2:
Ey = E1 + E2
Now, let's substitute the values and calculate the electric field:
Ex = (8.99 x 10^9 N*m^2/C^2) * (8.00 x 10^-9 C) / (12.0 m)^2 - (8.99 x 10^9 N*m^2/C^2) * (6.00 x 10^-9 C) / (9.0 + 0 m)^2
Ey = (8.99 x 10^9 N*m^2/C^2) * (8.00 x 10^-9 C) / (12.0 m)^2 + (8.99 x 10^9 N*m^2/C^2) * (6.00 x 10^-9 C) / (9.0 + 0 m)^2
Now you can calculate the values of Ex and Ey.