(Solved): An external E-field produces aligned electric dipole moments in a dielectric material; this in tur ...

An external $$E$$-field produces aligned electric dipole moments in a dielectric material; this in turn results in the appearance of excess charge on the surface of the material. The excess charge is not free to move around, so it is called "bound" charge, and its area density is $${?}_b$$. Consider a slab of dielectric filling the gap between the parallel plates of a capacitor. The charge on the plates would produce a field in the gap $$E_0$$ if the dielectric were not there. (a) Explain why the electric dipoles are aligned with the external field, and then show that the bound surface charge next to the positive plate of the capacitor is negative, and vice versa. [Draw pictures to illustrate your points.] (b) Express the net field in the slab, $$E_{\text{net }}$$, in terms of $$E_0$$ and $${?}_b$$. (c) Because the forces and torques that produce the aligned dipoles are proportional to $$E_{\text{net }}$$, we can write $${?}_b={?}_0?E_{\text{net }}$$, where the dimensionless number $$?$$ is the electric susceptibility of the material. Use this to show that $$E_{\text{net }}=\frac{E_0}{1+?}=\frac{E_0}{?}$$, where the dielectric constant $$?$$ is defined as $$?=1+?$$.