MHTCET 11 May 2024 Shift 1 Paper

Concept:When a dielectric fills one capacitor completely, its capacitance becomes $KC$, and the effective series capacitance changes from $\frac{C}{2}$.Explanation:Initially, two identical capacitors of capacitance $C$ are in series.Their effective capacitance is:$$C_{\text{initial}} = \frac{C}{2}.$$After filling one capacitor with a dielectric of constant $K$, its new capacitance becomes $KC$.The other capacitor still has capacitance $C$.For series combination of $KC$ and $C$, the new effective capacitance is:$$C' = \frac{(KC)(C)}{KC + C} = \frac{KC}{K + 1}.$$The change in effective capacitance is the difference between the new and initial values:$$\Delta C = \frac{KC}{K+1} - \frac{C}{2}.$$Taking the common denominator $2(K+1)$, we get:$$\Delta C = \frac{2KC - C(K+1)}{2(K+1)} = \frac{C(K-1)}{2(K+1)}.$$Therefore, the change in effective capacitance is:$$\Delta C = \frac{C}{2}\left(\frac{K-1}{K+1}\right).$$Answer:The correct option is C: $\frac{C}{2}\left[\frac{K-1}{K+1}\right]$.