Question

can you please me answer of this question
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Answer 1

1.) capacitors $\large\rm { C_{1} , C_{2} \ \sf{and} \ C_{3}}$ are connected in parallel. the same potential is applied accross all three capacitors.

Let $\large\rm { Q_{1} , Q_{2} \ \sf{and} \ Q_{3}}$ are chargea on the three plates of capacitors such that,

$\large\rm { Q_{1} = C_{1} V}$

$\large\rm { Q_{2} = C_{2} V}$

$\large\rm { Q_{3} = C_{3} V}$

Total charge stored is $\large\rm { Q = Q_{1} + Q_{2} + Q_{3}}$

and equivalent capacitance $\large\rm { C_{eq} = \frac{Q}{V} }$

so, $\large\rm { C_{eq} = C_{1} + C_{2} + C_{3}}$ ✓

2) Let 'C' is the capacitance of the capacitor. Let 'Q' and 'V' be the charge and potential difference respectively. Let 'q' be the charge on the capacitor at some stage during charging and 'v' is the corresponding potential difference between the plates.

So, $\large\rm { dw = V \ dq}$

but $\large\rm { \ C = \frac{q}{v}}$

so, $\large\rm { dw = \frac{q}{C} dq}$

Total work done in charging the capacitor is given by:

$\large\rm { w = \displaystyle\int \rm{dw }}$

$\large\rm { = \displaystyle\int_{0}^{\rm{Q}} \rm{\frac{qdq}{C} }}$

$\large\rm { \frac{1}{C} \cdot \bigg [ \frac{q^{2}}{2} \bigg ] _{0}^{Q} }$

This work done is stored in the form of Potential Energy, so energy of a charged condenser:

$\large\rm { u = \frac{Q^{2}}{2C}}$

but $\large\rm { \ Q = CV}$

so

$\large\rm { u = \frac{1}{2} QV}$