1. Verify if the following field can represent an electrostatic field in a given region.

If yes, find a potential function and determine the charge density at the point (2,3,0).

The curl of the field is zero and hence the field can be electrostatic. The potential function is . The  divergence of the field is given by  which is equal to . Thus at (2,3,0) the charge density is

2. The electric field in certain region of space is given in spherical polar coordinates as . Determine the charge density.

3. Find the electric field of a dipole on a point (a) along its axis and (b) along its perpendicular bisector in Cartesian coordinates.

We have shown that the field of an electric dipole at an angle  to the axis of the dipole is given by . Taking the axis along the z axis, along the axis so that the field is . Perpendicular to the axis (take it as x direction, , the field is

4. Show that the magnitude of the electric field of a dipole at an angle  to the axis of the dipole is given by .

This can be solved in a straightforward manner.

We have seen in the Tutorial problem 4 that the field at a perpendicular distance y from the edge of a semi-infinite line  has equal components perpendicular to the line and parallel to it (but in opposite direction), each component being  . In the figure below we show the field directions at P due to two semi-infinite wires, the directions shown in red is due to the line below the x-axis and that shown in blue is due to the wire above the x-axis.  The angle between A1P and B1P is given as 600 and distance of P from each of the wires is R.  Considering one of the   

wires  (say BN), we resolve the fields along x and y directions :

If we consider the wire AM, we get similar expressions but the y-components are in the reverse to direction to that due to BN while the x components add up.  The resulting contribution due to the two lines is

To complete the problem we have to add the contribution due to the 600 arc. We have seen  in Problem 3 that the field due the arc is along its axis which in this case is the x axis and has a magnitude given by . Adding  all contributions we have .