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.

A z-directional point dipole with moment P0 = 4πε0 is placed at the origin in empty space. As you know, the potential and electric field of this dipole are expressed in these formulas in spherical coordinates.Calculate the work to be done against the external electric field in order to move a point charge of Q = 1 Coulomb under the electric field effect created by the dipole from (9,3,0) to (3,9,0).

A point charge in the amount of Q is placed between an infinite grounded conducting wall to its left at a distance of L and a grounded conducting sphere of radius r to its right at a distance of R (R>r). The line connecting the center of the sphere and the point charge extends leftward to strike the wall at normal incidence. Find the electrostatic potential everywhere to left of the wall but outside the sphere.

A complex electrostatic potential is of the form: n(z) = Az + B, where A and B are real. Identify and sketch the equipotentials and lines of force of the given potential N(z).

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…

(a) What property of the Electric Field E allows us E can be expressed as a gradient of a scalar (V). (b) If you integrate Electric Field E along a path, say from point A to B, you get the potential difference between points B and A. If you integrate E along a closed loop what do you get?

An uncharged, infinitely long conducting cylinder of radius a is placed in an initially uniform clectric ficld E = Eoi, such that the cylinder's axis lics along the z axis. The resulting clectrostatic potential is V (x, y, z) = Vo for points inside the cylinder, and Ega?x x² + y? V (r, y, z) = Vo – Egx + for points outside the cylinder, where Vo is the (constant) clectrostatic potential on the conductor. Usc this expression to determine the resulting clectric ficld, E.

Compute for the work done, in millijoules, in moving a 2-nC charge from A(1, 3, 1) m to B(4, 4, -4) m against the electric field due to a disk charge of radius 3 m on the plane x = 0. The disk has a total charge of 6 mC.

An infinite grounded conducting plane is located at z = 0 and fills the x – y plane so that the electrostatic potential in the plane V (x, y, 0) is zero for all x and y. A single charge q is placed a distance a above the plane at the point (0,0, a). There are no other charges in the region z>0. (a) Using the boundary conditions and the first uniqueness theorem, find the electrostatic potential V (x, y, z) for all points above the plane: all x and y, and all z > 0. (b) Using the formula E = -V, calculate the electric field É also for all points above the plane. What is the electric field below the plane for z < 0? (c) Find the charge density per area o(x, y) for all x and y on the grounded plane, by using Gauss' law and the value of E just above the plane. (d) Integrate your result for o(x, y) to find the total amount of charge that is on the grounded plane wi charge q that was placed above the plane. This means that you should find the a circle of radius R, whose center is directly…

U:12) On a square wire frame of side length a in empty space, there is pi linear charge density. Calculate the value of the potential V = ke • Pi · A [Volt] at the height h = 4 • a on the strut exiting the center of this square frame. Find the numerical value of the value of A?