To solve this problem, we need to follow several steps carefully in order to compute the required quantities: the electric flux density (D) and the permittivity of the dielectric (ε).
First, let’s summarize the given data:
1. Potential difference (pd) = 10 kV
2. Area of each plate (A) = 10000 mm²
3. Distance between the plates (d) = 1 mm
4. Capacitance (C) = [tex]\(3 \times 10^{-4}\)[/tex] µF
Step 1: Convert all given values into SI units
- Potential difference (pd): 10 kV = 10000 V ([tex]\[1 kV = 1000 V\][/tex])
- Area (A): 10000 mm² = 0.01 m² ([tex]\[1 mm² = 1 \times 10^{-6} m²\][/tex])
- Distance (d): 1 mm = 0.001 m ([tex]\[1 mm = 1 \times 10^{-3} m\][/tex])
- Capacitance (C): [tex]\(3 \times 10^{-4}\)[/tex] µF = [tex]\(3 \times 10^{-10}\)[/tex] F ([tex]\[1 µF = 1 \times 10^{-6} F\][/tex])
Step 2: Calculate the electric charge (Q) on the plates
The charge Q stored in a capacitor is given by:
[tex]\[ Q = C \times V \][/tex]
where C is the capacitance and V is the voltage.
Given:
[tex]\[ C = 3 \times 10^{-10} \, F \][/tex]
[tex]\[ V = 10000 \, V \][/tex]
Therefore:
[tex]\[ Q = 3 \times 10^{-10} \times 10000 \][/tex]
[tex]\[ Q = 3 \times 10^{-6} \, C \][/tex]
Step 3: Calculate the electric flux density (D)
Electric flux density D is given by:
[tex]\[ D = \frac{Q}{A} \][/tex]
where Q is the charge and A is the area of the plate.
Given:
[tex]\[ Q = 3 \times 10^{-6} \, C \][/tex]
[tex]\[ A = 0.01 \, m² \][/tex]
Therefore:
[tex]\[ D = \frac{3 \times 10^{-6}}{0.01} \][/tex]
[tex]\[ D = 0.0003 \, C/m² \][/tex]
Step 4: Calculate the permittivity of the dielectric (ε)
The permittivity ε is given by:
[tex]\[ \epsilon = \frac{C \cdot d}{A} \][/tex]
where C is the capacitance, d is the distance between the plates, and A is the area of the plate.
Given:
[tex]\[ C = 3 \times 10^{-10} \, F \][/tex]
[tex]\[ d = 0.001 \, m \][/tex]
[tex]\[ A = 0.01 \, m² \][/tex]
Therefore:
[tex]\[ \epsilon = \frac{3 \times 10^{-10} \times 0.001}{0.01} \][/tex]
[tex]\[ \epsilon = \frac{3 \times 10^{-13}}{0.01} \][/tex]
[tex]\[ \epsilon = 3 \times 10^{-11} \, F/m \][/tex]
To summarize, the final results are:
- Electric flux density [tex]\(D = 0.0003 \, C/m²\)[/tex]
- Permittivity of the dielectric [tex]\(\epsilon = 3 \times 10^{-11} \, F/m\)[/tex]
These results match the numerical values calculated in our detailed solution.
