Question

A parallel plate capacitor has a parallel slab of copper inserted between and parallel to the two plates, without touching the plates. The capacity of the capacitor after the introduction of the copper sheet is:

• A. minimum when the copper slab touches one of the plates
• B. maximum when the copper slab touches one of the plates
• C. invariant for all positions of the slab between the plates
• D. greater than that before introducing the slab

Answer 1

Answer: (C), (D)

invariant for all positions of the slab between the plates
greater than that before introducing the slab

As the copper is a good conductor, the charge on the capacitor will appear on the large flat surfaces of the copper sheet, with the negative side of the copper facing the positive side of the capacitor. This arrangement can be considered to be two capacitors in series.
If thickness of the copper slab is $l$, the separation of plates between will be $\frac{1}{2}(d-l)$
The capacitance before insert slab is $C=\frac{A{ϵ}_0}{d}$
The capacitance of each capacitor after insert slab is $C^{\prime }=\frac{A{ϵ}_0}{(d-l)/2}$
Net capacitance after insert slab is $C_{eq}=\frac{C^{\prime }{C}^{\prime }}{C^{\prime }+C^{\prime }}=\frac{C^{\prime }}{2}=\frac{A{ϵ}_0}{(d-l)}$
thus, $C_{eq}>C$ and if sheet is very thin $(d>>l),C_{eq}=C$