- #1
roam
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Homework Statement
A dielectric slab with a susceptibility ##\chi_e## rests on a conducting plate whose upper surface carries a free surface charge density ##\sigma##. Show that the polarization surface charge density ##\sigma_{pol}## on the lower face of the dialectic slab is:
##\sigma_{pol} = - \sigma \frac{\chi_e}{1+\chi_e}##
Homework Equations
The polarization vector P is: ##P=\chi \epsilon_0 E##
The electric field between the surfaces is
##E=\frac{\sigma - \sigma_{pol}}{\epsilon_0} = \frac{\sigma}{\epsilon_0} \frac{1}{1+\chi}##
The Attempt at a Solution
I believe in this case ##\sigma_{pol} = P##, so combining the two equations I get
##\sigma_{pol}=P=\chi_e \epsilon_0 \frac{\sigma}{\epsilon_0} \frac{1}{1+\chi}##
##= \sigma \frac{\chi_e}{1+\chi_e}##
However I did not get the minus sign in front of my expression. What is wrong?
I know that ##\sigma_{pol}## and ##\sigma## must be of opposite sign, but mathematically how do I bring in the negative sign? Is my approach to the problem correct?
Any help would be appreciated.