Imagine you recently moved to Otaniemi with your best friend. Living together so far has been going pretty smoothly, but there is one issue that always makes you argue. Imagine a z-directed static electric dipole with dipole moment p, whose scalar p.ar potential is V(r) Your roommate started the fight because he claims that a point placed at a distance a at the x-axis (point A on the map) receives more electric field than a point at the same distance in the z-axis (point B). After all, point A is perpendicular to the static dipole. However, you argue that the magnitude of the electric field present in A is half that in B because the static dipole is z-polarized. 4 π € 0 R²* (a) (b) а N ….. PA р B A а > X +q -9 ---- ---- B A Your roommate is stubborn. Besides, he has been fooled by a toxic Youtube channel and doesn't believe in dipoles. Fortunately, he still believes in charge su- perposition. To convince him, you approximate the dipole with two point charges tq, separated by the distance d = a/N, being N a positive integer (N > 0). Moreover, qd = |p| (Figure 2 b). Find the expression of the electric field at points A and B in this case. --- d=a/N Check that the field directions are the same as for the case of the dipole. More- over, compare the magnitude of the field created by the two charges with the exact dipole. For example, how similar are the magnitudes if N = 3? If N increases (then d is reduced), do the magnitudes in A and B become similar to the static dipole or have the opposite effect?

Imagine you recently moved to Otaniemi with your best friend. Living together so far has been going pretty smoothly, but there is one issue that always makes you argue. Imagine a z-directed static electric dipole with dipole moment p, whose scalar p.ar potential is V(r) Your roommate started the fight because he claims that a point placed at a distance a at the x-axis (point A on the map) receives more electric field than a point at the same distance in the z-axis (point B). After all, point A is perpendicular to the static dipole. However, you argue that the magnitude of the electric field present in A is half that in B because the static dipole is z-polarized. 4 π € 0 R²* (a) (b) а N ….. PA р B A а > X +q -9 ---- ---- B A Your roommate is stubborn. Besides, he has been fooled by a toxic Youtube channel and doesn't believe in dipoles. Fortunately, he still believes in charge su- perposition. To convince him, you approximate the dipole with two point charges tq, separated by the distance d = a/N, being N a positive integer (N > 0). Moreover, qd = |p| (Figure 2 b). Find the expression of the electric field at points A and B in this case. --- d=a/N Check that the field directions are the same as for the case of the dipole. More- over, compare the magnitude of the field created by the two charges with the exact dipole. For example, how similar are the magnitudes if N = 3? If N increases (then d is reduced), do the magnitudes in A and B become similar to the static dipole or have the opposite effect?