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We know that due to external uniform fields, the dielectric atoms get polarized in the direction of the field, but what if the charge is inside the dielectric, the electric field of the charge is radial so will it be polarized upwards or downwards? In other words, will the bound charge density be positive downward or negative?

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As you have noted "dielectric atoms get polarized in the direction of the field". This means that if there is a radial field coming from the center of the dielectric it will be polarized outward from the center (there is no upwards or downwards in this problem, there is inwards and outwards). Think of the dielectric a collection of dipoles and each dipole aligns itself with the field. If there is a charge in the center, all the dipoles will point outward and on all the external perimeter of the dielectric there will be a bound charge (positive if the central charge is positive).

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