Step-by-step explanation:
Let's assume the charges are located at positions A and B, with the midpoint between them being the origin (0, 0, 0). Charge A is at (-d/2, 0, 0), and charge B is at (d/2, 0, 0).
The electric field due to charge A can be calculated as:
E_A = k * (q / R_A^2) * R_A_hat
where k is the electrostatic constant (k ≈ 9 x 10^9 N m^2/C^2), q is the magnitude of the charge, R_A is the distance from charge A to the point of interest, and R_A_hat is the unit vector pointing from charge A towards the point of interest.
Similarly, the electric field due to charge B is given by:
E_B = k * (q / R_B^2) * R_B_hat
where R_B is the distance from charge B to the point of interest, and R_B_hat is the unit vector pointing from charge B towards the point of interest.
Since the point of interest is at a distance z above the midpoint, its position vector is given by R_1 = (0, d/2, z).
To calculate the total electric field at this point, we sum the contributions from both charges:
E_total = E_A + E_B
Substituting the values of R_A, R_B, R_A_hat, and R_B_hat into the equations and simplifying, we can find the expression for the total electric field.
Now, let's consider the limit where z is much greater than d (z ≫ d). In this case, we can neglect the contributions from the horizontal components of the position vectors R_A and R_B since they become relatively small compared to the vertical component (z). Thus, the electric field simplifies to:
E_total ≈ 2 * k * (q / (d^2 + z^2)^(3/2)) * z_hat
where z_hat is the unit vector pointing in the positive z-direction.
Please note that the above expressions assume point charges and that the distances are much larger than the size of the charges. Additionally, the direction of the electric field is determined by the unit vectors R_A_hat, R_B_hat, and z_hat.