Final answer:
Adding a copper plate to a parallel plate capacitor changes the capacitance by splitting the capacitor into two series capacitors. The overall capacitance is found by summing the reciprocals of the individual capacitances due to the new distances d1 and d2, where d= d1 + d2.
Step-by-step explanation:
When a copper plate of thickness b is inserted into a parallel plate capacitor, the capacitance changes because the copper plate acts as a conductor that essentially splits the capacitor into two separate capacitors in series. Assuming that the copper plate does not completely fill the space between the plates, the original air gap d is now divided into two parts, d1 (the distance between one plate and the copper plate) and d2 (the distance between the copper plate and the other plate), with d1 + d2 equal to the original distance d. Each of these new gaps forms a separate capacitor with the copper plate. The overall capacitance of the system can be calculated by taking the reciprocal of the sum of the reciprocals of the capacitance of each individual capacitor (since capacitors in series have reciprocals that add up).
The capacitance of a parallel plate capacitor with air or vacuum between the plates can be found using the equation C = ε₀(A/d), where ε₀ is the permittivity of free space (8.85 × 10-12 F/m), A is the area of the plate in square meters, and d is the separation between the plates in meters. However, when inserting a conductive material like copper, we would need to account for its effects on the electric field distribution and the fact that charges will redistribute. Thus, inserting a copper plate effectively changes the configuration and requires more detailed considerations of the internal electric field and potential characteristics of both the copper and the surrounding dielectric.