Dielectric Effect Simulator

Physics Electricity and Magnetism • Electric Potential and Capacitance

Written by STEM Calculators Team Published January 12, 2026 Updated February 24, 2026

5. Dielectric Effect Simulator

Model capacitance increase with a dielectric: $C=\kappa C_0$ , electric energy density: $u=\tfrac12\,\kappa\varepsilon_0 E^2$ , plus polarization intuition and breakdown warning.

Inputs support: pi, e, sqrt(), sin, cos, exp, log. Use * for multiplication.

Mode + Inputs

Base capacitor: Vacuum/air capacitance $C_0$ (pF): Plate area $A$ (m²): Plate gap $d$ (m): $\varepsilon_0$ (F/m): $\kappa$ (material preset): Relative permittivity $\kappa$: Insertion model: Area fraction $f\in[0,1]$: Dielectric thickness $t$ (m): Electrical condition: Voltage $V$ (V): Charge $Q$ (C): Polarization animation: Breakdown preset: Breakdown field $E_{bd}$ (V/m):

Tip: “Partial insertion” models are approximations. Use Full for the classic $C=\kappa C_0$ case.

Ready

Steps

Enter values and click Solve.

Polarization + field intuition

Parallel-plate view (pan/zoom)

Hover: …

Dipoles align with the field; bound charges appear on the dielectric faces. This is a qualitative sketch (not a full EM solver).

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Theory — Dielectric Effect Simulator

A dielectric is an insulator whose molecules polarize in an electric field. Polarization reduces the effective field inside the material (compared with vacuum), so a capacitor can store more charge for the same applied voltage — i.e., its capacitance increases.

Capacitance increase

\[ C = \kappa\,C_0 \]

where $C_0$ is the vacuum/air capacitance and $\kappa=\varepsilon_r\ge 1$ is the relative permittivity.

Parallel-plate base formula

\[ C_0=\varepsilon_0\,\frac{A}{d}, \qquad C=\kappa\varepsilon_0\,\frac{A}{d} \]

Here $A$ is plate area, $d$ is separation, and $\varepsilon_0$ is the permittivity of free space.

Energy and energy density

The energy stored in a capacitor depends on what is held fixed:

\[ U=\frac12CV^2 \qquad(\text{fixed }V) \] \[ U=\frac{Q^2}{2C} \qquad(\text{fixed }Q) \]

For a (roughly) uniform field, the electric energy density is:

\[ u=\frac12\,\kappa\varepsilon_0E^2 \]

In a parallel-plate estimate, $E\approx V/d$. Real fields near edges are not perfectly uniform.

Polarization and bound charge (intuition)

Dipoles align with the field, producing bound surface charge on the dielectric faces. A simple linear-material preview uses:

\[ P=\varepsilon_0(\kappa-1)E \]

The simulator uses $P$ to drive a qualitative “polarization arrows” animation. This is not a full field solver — it’s meant to build intuition.

Partial insertion (approximate models)

When a dielectric is only partially inserted, the exact result depends on geometry. Two common approximations are:

1) Partial area (parallel regions):

\[ C \approx C_0\big((1-f)+\kappa f\big) \]

Here $f\in[0,1]$ is the fraction of plate area filled with dielectric (side-by-side regions).

2) Slab thickness in the gap (series layers):

\[ C \approx C_0\,\frac{d}{(d-t)+t/\kappa} \]

Here $t\le d$ is the dielectric thickness filling part of the gap along the field direction.

Breakdown warning

Real dielectrics have a maximum sustainable field (dielectric strength). If $E$ exceeds that, breakdown can occur. The simulator compares $|E|/E_{bd}$ using typical textbook values. This is only an estimate.

\[ \text{warning ratio}=\frac{|E|}{E_{bd}} \]

For university-level work, dielectric behavior can be frequency-dependent and lossy, and accurate modeling often requires constitutive relations, boundary conditions, and numerical methods (FEM).

Steps

Polarization + field intuition

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