Step 1: Understanding the Concept:
Energy density (\(u\)) is defined as the amount of electrostatic potential energy stored per unit volume in the electric field between the capacitor plates.
Unlike the total energy which depends on the size of the capacitor, the energy density depends only on the strength of the electric field (\(E\)) and the permittivity of the medium (\(\epsilon_0\)).
Step 2: Key Formula or Approach:
The energy density formula is:
\[ u = \frac{1}{2} \epsilon_0 E^2 \]
In a parallel plate capacitor, the electric field \(E\) is related to the potential difference \(V\) and the separation \(d\) as:
\[ E = \frac{V}{d} \]
Permittivity of free space \(\epsilon_0 \approx 8.854 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)\).
Step 3: Detailed Explanation:
Given values:
Capacitance, \(C = 1 \text{ \(\mu\)F}\) (Note: This value is not needed if we use \(V\) and \(d\)).
Potential difference, \(V = 20 \text{ V}\)
Separation, \(d = 1 \text{ \(\mu\)m} = 1 \times 10^{-6} \text{ m}\)
First, calculate the electric field \(E\):
\[ E = \frac{V}{d} = \frac{20}{1 \times 10^{-6}} = 20 \times 10^6 = 2 \times 10^7 \text{ V/m} \]
Now, calculate the energy density \(u\):
\[ u = \frac{1}{2} \times \epsilon_0 \times E^2 \]
\[ u = \frac{1}{2} \times (8.854 \times 10^{-12}) \times (2 \times 10^7)^2 \]
\[ u = 0.5 \times 8.854 \times 10^{-12} \times 4 \times 10^{14} \]
\[ u = 2 \times 8.854 \times 10^{-12 + 14} \]
\[ u = 17.708 \times 10^2 \]
\[ u = 1770.8 \text{ J/m}^3 \]
Rounding to significant figures or looking at the options:
\[ u \approx 1.8 \times 10^3 \text{ J/m}^3 \]
Step 4: Final Answer:
The energy density is approximately \(1.8 \times 10^3 \text{ J/m}^3\).