Consider the line dipole made of aligned line sink and source of equal strengths m. Let the sink be located at (x,y)=(0,0) and the source at (x,y)=(δ,0) . Using calculations analogous to those for the point dipole, show that, at large distances from the dipole, it’s potential can be approximated as Φ=−μ/r cosφ, where {r,φ} are the polar coordinates, and μ=mδ is called the line dipole moment. Then show that in the vector form Φ=−μ⋅r / r^2, where the dipole moment vector μ=μed , where ed is the unit vector pointing along the line connecting the sink with the source, and is the position vector originating at the dipole.

Use the dipole approximation to find what the potential (magnitude and sign) would be at point A and point B

The electric potential at large distances from the dipole (i.e. ligand >> d) is described by the relationship V (r, α) = kQ? ∙ cosα?2, where d is the distance between the charges of the dipole and r is the distance from the dipole center. In the Cartesian system whose beginning is halfway between the charges of the dipole, and the axis of the dipole is the X axis (see figure), and limited to the two-dimensional case, the given relationship is as follows:V (x, y) = k ∙ Q? ∙ x / (x2 + y2) 3/2Using this relationship, determine the coordinates of the electric field intensity vector at the point with the coordinates P = (R, R), i.e. at any point lying on the straight line lying at the angle α = π / 4 at a distance of √2 ∙ R from the origin of the coordinate system, at a large distance from the origin coordinate system.

Compute for the work done, in millijoules, in moving a 8-nC charge from A(0, 0, 1) m to B(0, 0, 7) m against the electric field due to a ring charge of radius 9 m on the plane z = 0 centered at the origin. The ring has a total charge of 8 mC.