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Found in: Page 699

### College Physics (Urone)

Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000

# Show that for a given dielectric material the maximum energy a parallel plate capacitor can store is directly proportional to the volume of dielectric (Volume=A.d. Note that the applied voltage is limited by the dielectric strength.

The strength saved with inside the parallel plate capacitor is without delay proportional to the volume of the dielectric.

$${U_E} \propto V$$

See the step by step solution

## Step 1: Definition and Capacitance

Capacitance: The proportion of a system's charge change to its corresponding potential change.

## Step 2: Concepts and Principles

The following is the amount of energy held in a capacitor with capacitance $$C$$ that is charged to a potential difference $$\Delta V$$: $${U_E} = \frac{1}{2}C{(\Delta V)^2}...(1)$$.

The parallel plate capacitor's capacitance is $$C = \kappa \frac{{{\varepsilon _0}A}}{d}...(2)$$

Where $$A$$ is the area of each plate, $${\rm{d}}$$is the distance between them, and $${\varepsilon _0}$$is the vacuum permittivity

\begin{aligned}{c}{\varepsilon _0} &= 1/(4\pi k)\\ &= 8.854 \times {10^{ - 12}}{\rm{ }}{C^2}/\left( {N \times {m^2}} \right)\end{aligned}

Any pair of conductors separated by an insulating substance is referred to as a capacitor. When the capacitor is charged, the two conductors have charges of equal magnitude $$Q$$ and opposite sign, and the positively charged conductor's potential $$\Delta V$$ with respect to the negatively charged conductor is proportional to $$Q$$ The ratio of $$Q$$ to $$\Delta V$$ determines the capacitance $$C$$

$$C = \frac{Q}{{\Delta V}}$$

And the potential difference between two points separated by a distance $$d$$ in a uniform electric field of magnitude $$E$$ is $$\Delta V = Ed...(3)$$.

## Step 3: Finding Capacitance

The energy stored in the parallel plate capacitor can be evaluated from Equation $$(1)$$:

$${U_E} = \frac{1}{2}C{(\Delta V)^2}$$

Where $$C$$ the capacitance of the parallel plate capacitor is found from Equation $$(2)$$ and $$\Delta V$$ is the potential difference across the capacitor found from Equation $$(3)$$:

Where $$Ad$$ is the volume $$V$$of the dielectric:

The term in parenthesis is constant; therefore, the energy stored in the parallel plate capacitor is directly proportional to the volume $$V$$of the dielectric:

$${U_E} \propto V$$