(a) Calculation of Dielectric Constant
Step 1 (Formula & Given Data): [0.5 Mark]
The capacitance of a parallel plate capacitor with a dielectric medium is given by $C_m = K \cdot C_0$, where $C_0$ is the capacitance in vacuum/air and $K$ is the dielectric constant.
Given: $C_0 = 10\text{ }\mu\text{F}$, $C_m = 80\text{ }\mu\text{F}$.
Step 2 (Mathematical Working): [0.5 Mark]
$$K = \frac{C_m}{C_0}$$ $$K = \frac{80\text{ }\mu\text{F}}{10\text{ }\mu\text{F}}$$Step 3 (Final Answer): [0.5 Mark]
The dielectric constant of the medium is $K = 8$ (Dimensionless).
(b) Equivalent Capacitance in Series
Step 1 (Governing Law): [0.5 Mark]
For $n$ capacitors connected in series, the reciprocal of the equivalent capacitance ($C_{eq}$) is the sum of the reciprocals of individual capacitances.
Step 2 (Mathematical Working): [0.5 Mark]
$$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}$$ Since all $n$ capacitors have equal capacitance $C$: $$\frac{1}{C_{eq}} = \underbrace{\frac{1}{C} + \frac{1}{C} + \dots + \frac{1}{C}}_{n \text{ times}} = \frac{n}{C}$$Step 3 (Final Answer): [0.5 Mark]
The equivalent capacitance is $C_{eq} = \frac{C}{n}$.
(c) Effect of Dielectric on Potential and Energy
Step 1 (Governing Law): [0.5 Mark]
When the battery is disconnected, the charge ($Q$) on the plates remains constant. The new capacitance becomes $C' = K \cdot C$.
Step 2 (Mathematical Working): [1.0 Mark]
- Potential Difference ($V$): Since $V = \frac{Q}{C}$, and $C$ increases to $KC$, the new potential $V' = \frac{Q}{KC} = \frac{V}{K}$. Thus, the potential difference decreases by a factor of $K$.
- Energy Stored ($U$): Using $U = \frac{Q^2}{2C}$, the new energy $U' = \frac{Q^2}{2(KC)} = \frac{U}{K}$. Thus, the energy stored decreases by a factor of $K$.
Step 3 (Final Conclusion): [0.5 Mark]
Both potential difference and energy stored decrease by a factor of $K$ because the charge is trapped and cannot be replenished, while the capacity to hold charge increases.
💡 EL10 Concept Insight: Think of the capacitor like a bucket; adding a dielectric is like making the bucket wider, so even with the same amount of water (charge), the water level (potential) drops,