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The field produced by charges in a slab of finite thickness “d” This section shows that the electric field produced by an infinitesimally-thin plate, at point along the central axis and far away from plane, is independent of the position of the plate. [Basically, the justification argument is that if the sheet is shifted to the right, then the separation between the sheet at the observation point decreases, hence given rise to a change in phase of the field arriving at P. But, the field at the sheet would have also been shifted. These two phase shift compensate. This argument is detailed below]. Consider two plates, one located at z = 0 and having 1 charges per unit area and another at z = having 2 charges per unit area. Charges on these plates will be excited by the incident external field. We wish to calculate the electric field produced by the plate’s oscillating charges at a point P. Y 2 1 External incident electric field P Einc ( z,t ) Eo ei(ωt - kz ) z Z z At z = 0 the electric field changes with time as E(0,t) Eoeiωt At z = the electric field changes with time as E( ,t) e - ik E eiωt o Since the local charges at each plate respond to the local electric fields, the acceleration of the charges at each plate will have, then, similar phase differences At z = 0 the acceleration of the charges vary with time as a(0,t ) a eiωt o At z = the acceleration of the charges vary with time as a( ,t ) e- ik a eiωt o The charges at the different plates will create an electric field at P(0, 0, z) (1) We will assume that the charges on the plate at the right are excited by the incident radiation that did not interacted with the charges on the left. We also arbitrarily discount that the fields produced by the charges on the left plate do not excite the charges on the right plate.) The external fields put in motion the charges on the plates Motion of the charges at z = 0 x(t) xo (ω) eiωt v(t) ix (ω) eiωt o Motion of the charges at z = x(t) xo (ω) e - ikeiωt v(t) ix (ω) e - ikeiωt o The charges in motion create an electric field at point P(0, 0, z) Contribution from the Plate-1 to the field at P E1 ( z, t ) -1 q 2 c i xoei (t - z/c ) (2) o which was basically obtained from the following calculation ~ [acceleration of one charge ] x [ a value resulting when ] at plate-1 integrating the contribution from many charges ~ [ acceleration at plate -1 ] But evaluated at a time before t [ i (c/w) ] How long before? Answer: a time t = distance/c where “distance” is the separation between the plate and the point pf observation For the case of sheet-1: “distance” =z ~[ eit ]at t- z/c ~[ eit-z/c) [ i (c/w) ] ] [ i (c/w) ] Plate-1 Here z is the distance from the point P to the plate-1. Contribution to the field at P from the Plate-2 E2(z,t) ~ [acceleration of one charge ] x [ a value resulting when ] at plate-2 integrating the contribution from many charges (3) ~ [ acceleration at plate -2 ] But evaluated some time before t [ i (c/w) ] How long before? Answer: a time t = distance/c where “distance” is the separation between the plate and the point pf observation. For the case of sheet-2: “distance” = z- ~ [ e - ik eiωt ] at t - (z - ) /c [ ~ [ e - ik eiω [t - ( z - )/c] i (c/w) ] ] [ i (c/w) ] Using k= /c ~ [ eiωt e -i z/c ] [ i (c/w) ] plate-2 (4) Notice, one obtains the same result expressed in (3) for the plate-1. E2 ( z,t ) - 2 q 2 c [ i xo (ω) eiω( t - z/c )] Independent of (5) o Contribution to the field at P from the sheet-2 That is, from (4) and (5), the electric field produced by a plate is independent of the position of the plate along the z-axis. The main conclusion we draw from the above analysis is that, the net effect of a sheet of charges on the point of observation “P” is to produce a field component that is /2 lagging the field produced by the source. No matter where we place the sheet, the net effect is to produce a field that is /2 lagging the field produced by the source. The above result implies that if we had not at infinitesimal-thin plate, but a thick plate of thickness “d”, the total electric field at P will have the form obtained for an infinitesimally-thin plate, except that instead of we would have to input Nod (where No is the number of charges per unit volume, at the center of the plate. No decreases away from the center.) E due to a plate ( z,t ) - N o d of thickness d External incident electric field Einc(z,t) Eoei(ωt - kz ) q 2 c [ i xo (ω) eiω( t - z/c )] o Y xo() d P z Z