Solution: Heat Dissipated in Conducting Plate
Physics Principle: The heat dissipated corresponds to the electrostatic potential energy stored in the induced charge distribution. This energy is equivalent to the energy of the electric field “excluded” from the volume of the conductor.
Calculation:
The energy density of the electric field is $u = \frac{1}{2}\epsilon_0 E^2$.
When the conductor is placed in the field, the field inside its volume becomes zero due to polarization. The energy “stored” in this polarization is equal to the field energy that would exist in that volume in free space.
Volume of the plate = $Area \times thickness = A \cdot d$.
$$ \text{Heat Dissipated} = \text{Energy Density} \times \text{Volume} $$ $$ H = \left( \frac{1}{2}\epsilon_0 E^2 \right) (Ad) $$
$$ H = \frac{1}{2}\epsilon_0 E^2 Ad $$