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 diectric 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 dielectric 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

Battery Connection in Capacitor Circuits

When a capacitor remains connected to a battery, the voltage across the capacitor is maintained constant. This forces any changes in capacitance, such as those induced by introducing a dielectric, to be accompanied by a corresponding change in the stored charge. The battery supplies additional charge to accommodate the increased capacitance, which contrasts with situations where the capacitor is isolated, resulting in different energy and charge dynamics.

Energy Storage in Capacitors

The energy stored in a capacitor is given by formulas such as U = 1/2 CV^2 when the voltage is constant, highlighting that energy depends on both the capacitance and the voltage applied. An increase in capacitance due to the dielectric, when the voltage is held constant, leads to an increase in stored energy. Understanding this relationship is key to explaining how the energy changes when a dielectric is introduced.

Parallel-Plate Capacitor

This is a simple capacitor configuration composed of two parallel conductive plates separated by a distance. Its capacitance in vacuum or air is determined by the area of the plates and the distance between them. Understanding the basic structure and mathematical formulation of a parallel-plate capacitor is foundational to analyzing how modifications, such as introducing a dielectric material, affect its behavior.

Dielectric Materials

Dielectrics are insulating materials inserted between the plates of a capacitor. They become polarized in the presence of an electric field, reducing the effective field between the plates and increasing the overall capacitance. The dielectric constant (?) is a measure of how much the material increases the capacitance compared to a vacuum, and it plays a crucial role in altering both the stored energy and charge distribution when inserted into a capacitor.

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