Formula: Plate Capacitor Energy Volume Electric field (E field)

$$W_{\text{e}} ~=~ \frac{1}{2} \, \varepsilon_0 \, \varepsilon_{\text r} \, V \, \class{purple}{E}^2$$ $$W_{\text{e}} ~=~ \frac{1}{2} \, \varepsilon_0 \, \varepsilon_{\text r} \, V \, \class{purple}{E}^2$$ $$V ~=~ \frac{ 2 W_{\text{e}} }{ \varepsilon_0 \, \varepsilon_{\text r} \, \class{purple}{E}^2 }$$ $$\class{purple}{E} ~=~ \sqrt{ \frac{2 \, W_{\text{e}} }{\varepsilon_0 \, \varepsilon_{\text r} \, V} }$$ $$\varepsilon_{\text r} ~=~ \frac{ 2 W_{\text{e}} }{ \varepsilon_0 \, V \, \class{purple}{E}^2 }$$

Rearrange formula

Electrical energy

$$ W_{\text{e}} $$ Unit $$ \mathrm{J} = \mathrm{Nm} = \frac{ \mathrm{kg} \, \mathrm{m^2} }{ \mathrm{s}^2 } $$

The energy that is 'stored' in the plate capacitor. The energy is stored in the electric field between the plates.

Volume

$$ V $$ Unit $$ \mathrm{m}^3 $$

The volume enclosed by the two capacitor plates.

Electric field

$$ \class{purple}{\boldsymbol E} $$ Unit $$ \frac{\mathrm{V}}{\mathrm{m}} = \frac{\mathrm{N}}{\mathrm{C}} = \frac{\mathrm{kg} \, \mathrm{m}}{\mathrm{A} \, \mathrm{s}^3} $$

The E-field indicates how large the electric force on a test charge would be if it were placed at a specific location between the capacitor plates.

Relative permittivity

$$ \varepsilon_{\text r} $$ Unit $$ - $$

Relative permittivity is a dimensionless quantity that characterizes the dielectric (non-conductive material) between the capacitor plates. Depending on the material used, the dielectric can be used to effect the capacitance of the capacitor, for example.

In vacuum, the permittivity by definition has the value $ \varepsilon_{\text r} = 1 $. The water has approximately a value of $ \varepsilon_{\text r} \approx 80 $.

Vacuum Permittivity

$$ \varepsilon_0 $$ Unit $$ \frac{\mathrm{As}}{\mathrm{Vm}} $$

The vacuum permittivity is a physical constant that appears in equations involving electromagnetic fields. It has the following experimentally determined value: $$ \varepsilon_0 ~\approx~ 8.854 \, 187 \, 8128 ~\cdot~ 10^{-12} \, \frac{\mathrm{As}}{\mathrm{Vm}} $$

Related formulas

Plate Capacitor (Force, Voltage, Charge, Distance)

$$ F ~=~ \class{red}{q} \, \frac{U}{d} $$

Plate Capacitor (Capacitance)

$$ C ~=~ \varepsilon_0 \, \varepsilon_{\text r} \, \frac{A}{d} $$

Plate Capacitor (Electric Field, Voltage, Distance)

$$ \class{purple}{E} ~=~ \frac{U}{d} $$

Capacitor (Energy, Capacitance, Charge)

$$ W_{\text{e}} ~=~ \frac{1}{2} \, \frac{Q^2}{C} $$

Capacitor (Energy, Voltage, Capacitance)

$$ W_{\text e} ~=~ \frac{1}{2} \, C \, U^2 $$

Plate capacitor (potential, voltage, distance)

$$ \varphi_x = - \frac{U}{d} \, x ~+~ \varphi_1 $$

Plate Capacitor (Attraction Force, Voltage)

$$ F ~=~ \frac{\varepsilon_0 \, \varepsilon_{\text r} \, A}{2d^2}\, U^2 $$

Plate Capacitor (Attraction Force, Charge)

$$ F ~=~ \frac{Q^2}{2\varepsilon_0 \, \varepsilon_{\text r} A} $$

Plate Capacitor: Voltage, Capacitance and Eletric Force

Here the plate capacitor is simply explained. With the help of the capacitor you will learn about the electric voltage, electric field and electric capacitance and how to calculate them for a plate capacitor.

Derivations & Experiments

Plate Capacitor: Potential, E-field, Charge and Capacitance

Here, the electrostatic potential, electric field and the capacitance of a plate capacitor are derived using Laplace's equation.

Exercises with solutions

Create a Plate Capacitor with a Certain Capacitance

In this exercise (with solution) you have to calculate the distance between the capacitor plates for a given capacitance and and than use relative permittivity.

Charge in a thundercloud

The exercise teaches you how to calculate the charge of a thundercloud using the capacitance and E-field in the plate capacitor.

Capacitor with and without dielectric in comparison

In this exercise with solution about the plate capacitor, the capacitance, voltage and energy must be compared with and without dielectric.

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