A A 12.5 μF capacitor is connected to a power supply that...

A $\mathrm{A} 12.5 \mu \mathrm{F}$ capacitor is connected to a power supply that keeps a constant potential difference of 24.0 $\mathrm{V}$ across the plates. A piece of material having a dielectric constant of 3.75 is placed between the plates, completely filling the space between them. (a) How much energy is stored in the capacitor before and after the dielectric is inserted? (b) By how much did the energy change during the insertion? Did it increase or decrease?

Key Concepts

Capacitance

Capacitance is a measure of a capacitor's ability to store electric charge per unit voltage. It is defined by the relationship C = Q/V, where Q is the charge stored on the capacitor and V is the voltage across its plates. This property determines how much energy a capacitor can store and is a fundamental parameter in circuit design and analysis.

Dielectric Materials

Dielectric materials are insulators that, when inserted between the plates of a capacitor, increase its capacitance. This increase occurs because the dielectric reduces the effective electric field between the plates, allowing more charge to be stored for the same applied voltage. The effectiveness of a dielectric material is quantified by its dielectric constant, which is the ratio of the capacitance with the dielectric to the capacitance in vacuum.

Energy Stored in Capacitors

The energy stored in a capacitor is given by the equation U = ½CV², where C is the capacitance and V is the voltage across the capacitor. This expression shows that the stored energy depends on the capacitor's ability to hold charge (its capacitance) and the square of the voltage. Understanding how energy storage is affected by variations in capacitance is key when modifying capacitor systems.

Effect of Dielectric Insertion Under Constant Voltage

When a capacitor is connected to a constant voltage source, inserting a dielectric increases the capacitance by a factor equal to the dielectric constant. Because the voltage remains fixed, this increase in capacitance leads directly to an increase in the stored energy, as evidenced by the energy formula U = ½CV². This concept is important for understanding how capacitor performance improves with_dielectric materials and how energy changes during such modifications.

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