(a) How many excess electrons must be added to one plate and...

(a) How many excess electrons must be added to one plate and removed from the other to give a 5.00 $\mathrm{nF}$ parallel-plate capacitor 25.0$\mu \mathrm{J}$ of stored energy? (b) How could you modify the geometry of this capacitor to get it to store 50.0$\mu$ J of energy without changing the charge on its plates?

Key Concepts

Energy Storage in Capacitors

This concept involves understanding that the energy stored in a capacitor can be expressed in multiple equivalent forms such as U = ½CV² and U = Q²/(2C), where U is the energy, C is the capacitance, V is the voltage, and Q is the charge. This relation is key to determining how much charge (or number of excess electrons) is needed for a desired energy storage and how energy storage changes with adjustments to capacitance or charge.

Capacitance of a Parallel-Plate Capacitor

The capacitance of a parallel-plate capacitor is determined by its physical geometry and the properties of the medium between the plates. Specifically, it is given by the equation C = ??A/d for a capacitor in a vacuum (or adjusted by the dielectric constant for other media), where A is the area of the plates, d is the separation between them, and ?? is the permittivity of free space. This concept is essential for understanding how altering the plate area or separation modifies the capacitance.

Quantization of Charge

Charge quantization is the principle stating that charge exists in discrete amounts, with the elementary charge (the magnitude of the charge of an electron) being the smallest unit. In problems involving capacitors, calculations often require converting a total charge into the equivalent number of electrons or vice versa, using the known value of the elementary charge.

Impact of Geometry on Energy for Fixed Charge

When the charge on a capacitor is kept constant, the stored energy is inversely proportional to the capacitance as indicated by U = Q²/(2C). This means that modifications to the capacitor’s geometry, which alter its capacitance (for example by changing the plate separation or area), can directly affect the energy stored. Understanding this relationship is crucial when adjusting the physical structure of a capacitor to achieve a desired energy storage without altering its charge.

Transcript

Please give Ace some feedback