1 Answer
The formula for displacement current in a dielectric medium is derived from Maxwell's equations, specifically the modified Ampère's Law, which includes the displacement current term. The displacement current \( I_d \) is related to the rate of change of the electric field in a dielectric material.
The displacement current density \( \mathbf{J}_d \) is given by:
\[
\mathbf{J}_d = \epsilon \frac{\partial \mathbf{E}}{\partial t}
\]
Where:
Now, the total displacement current \( I_d \) through a surface is obtained by integrating the displacement current density over the area \( A \) through which it passes:
\[
I_d = \int_A \mathbf{J}_d \cdot d\mathbf{A} = \epsilon \int_A \frac{\partial \mathbf{E}}{\partial t} \cdot d\mathbf{A}
\]
In simpler terms:
This concept is important in scenarios involving capacitors, where the electric field between the plates changes as the capacitor charges or discharges.
The displacement current density \( \mathbf{J}_d \) is given by:
\[
\mathbf{J}_d = \epsilon \frac{\partial \mathbf{E}}{\partial t}
\]
Where:
- \( \mathbf{J}_d \) is the displacement current density (A/m²)
- \( \epsilon \) is the permittivity of the dielectric material (F/m)
- \( \frac{\partial \mathbf{E}}{\partial t} \) is the time rate of change of the electric field (V/m·s)
Now, the total displacement current \( I_d \) through a surface is obtained by integrating the displacement current density over the area \( A \) through which it passes:
\[
I_d = \int_A \mathbf{J}_d \cdot d\mathbf{A} = \epsilon \int_A \frac{\partial \mathbf{E}}{\partial t} \cdot d\mathbf{A}
\]
In simpler terms:
- The displacement current is proportional to how fast the electric field changes over time and the permittivity of the dielectric material.
- The integral accounts for the area over which this displacement current is flowing, and it gives you the total current in the dielectric.
This concept is important in scenarios involving capacitors, where the electric field between the plates changes as the capacitor charges or discharges.