A parallel-plate capacitor of plate separation d is charged...

A parallel-plate capacitor of plate separation $d$ is charged to a potential difference $\Delta V_{0}$. A dielectric slab of thickness $d$ and dielectric constant $\kappa$ is introduced between the plates while the battery remains connected to the plates. (a) Show that the ratio of energy stored after the diclectric is introduced to the energy stored in the empty capacitor is $U / U_{0}=\kappa .$ Give a physical explanation for this increase in stored energy. (b) What happens to the charge on the capacitor? (Note that this situation is not the same as in Example 26.7 , in which the battery was removed from the circuit before the dielectric was introduced.)

Key Concepts

Constant Voltage Supply (Battery Connection)

When a capacitor is connected to a battery, the voltage across it is fixed by the battery. This connection means that any change in the capacitance, such as by introducing a dielectric material, requires additional charge to flow to the capacitor to maintain the same voltage. This results in a modification of both the stored energy and the amount of charge on the capacitor.

Charge Redistribution and Energy Increase

The increase in capacitance due to the dielectric causes more charge to be drawn from the battery to maintain the fixed voltage. The additional charge not only increases the energy stored in the capacitor (as indicated by the U = 1/2 CV^2 relationship) but also explains the physical mechanism behind the observed increase in energy when a dielectric is introduced in a constant-voltage scenario.

Stored Energy in a Capacitor

The energy stored in a capacitor is given by the formula U = 1/2 CV^2 when the voltage is held constant. Inserting a dielectric increases the capacitance; when the voltage remains unchanged (as with a battery connected), this directly leads to an increase in the energy stored, since energy grows proportionally with capacitance under constant voltage conditions.

Parallel-Plate Capacitor

A parallel-plate capacitor is a device consisting of two conductive plates separated by an insulating gap. Its capacitance is determined by the area of the plates, the distance between them, and the permittivity of the material between the plates. This device stores energy in the electric field established by a potential difference across the plates.

Dielectric Material

A dielectric material is an insulating substance inserted between the plates of a capacitor. Its key property is the dielectric constant (?), which measures how effectively the material can be polarized by an electric field. This polarization leads to a reduction in the effective field in the gap, thereby increasing the capacitance of the capacitor by a factor of ?.

Please give Ace some feedback