Exercise:
A po dipole is an idealisation where we ase that the two charges are arbitrarily close together i.e. that the distance d between them is much smaller than the distance r to a po in the field. In this case the electric field vector is given by vecEvec r frack_Cr^ left leftvecp hatrright hatr - vecpright where vecpq vecd is the dipole moment vecr the displacement vector from the centre of the dipole to the po in the field and hatr a unit vector in the direction of vecr i.e. hatrvecr/r. abcliste abc Derive the expressions for the electric field at a po on the line defined by the direction of the dipole i.e. vecr parallel vecp and for the field at a po perpicular to the dipole i.e. vecr perp vecp. abc Calculate the components of the electric field produced by a water molecule dipole moment ppWO for a distance rO and an angle thO between the direction of the dipole moment and the displacement vector. abcliste

Solution:
abcliste abc For a po on the line defined by the direction of the dipole we know that vecp hatr p and p hatr vecp It follows for the electric field vector vecE frack_Cr^ left vecp-vecp right EparF For a po at a right angle to the dipole the dot product is zero. Therefore the electric field vector is vecE EperpF abc We ase that the dipole moment is oriented in the positive x direction: vecp pmatrix p pmatrix The unit vector hatr can then be written as hatr fracvecrhatr fracr pmatrix cosvartheta sinvartheta pmatrixr pmatrix cosvartheta sinvartheta pmatrix The dot product is then just vecp hatr p_x hatr_x + p_y hatr_y + p_z hatr_z p cosvartheta It follows for the components of the field vector E_x frack_Cr^ left p cosvartheta hatr_x - p_x right frack_Cr^ left p cosvartheta hatr_x - p right ExF frack times pWleftrright^ times left times cos^th - right Ex approx resultExP E_y frack_Cr^ left p cosvartheta hatr_y - p_y right frack_Cr^ left p cosvartheta hatr_y - right EyF frack times pWleftrright^ times left times costh times sinth right Ey approx resultEyP E_z frack_Cr^ left p cosvartheta hatr_z - p_z right frack_Cr^ left p cosvartheta - right abcliste