Question

plz find the equivalent capacitance between A and B of iii and iv

Answer 1

To Find :

▪ Equivalent capacitance between A and B.

Formula :

✏ For series connection :

$\bigstar\:\underline{\boxed{\bf{\red{\dfrac{1}{C_s}=\dfrac{1}{C_1}+\dfrac{1}{C_2}}}}}$

✏ For parallel connection :

$\bigstar\:\underline{\boxed{\bf{\blue{C_p=C_1+C_2}}}}$

Calculation :

⚾ Answer-iii) :

↗ Equivalent of part A (series) :

$\implies\sf\:\dfrac{1}{C_1}=\dfrac{1}{3C}+\dfrac{1}{3C}+\dfrac{1}{3C}\\ \\ \implies\sf\:\dfrac{1}{C_1}=\dfrac{3}{3C}\\ \\ \implies\bf\:C_1=C$

↗ Equivalent of part B (parallel) :

$\implies\sf\:C_2=C_1+C\\ \\ \implies\sf\:C_2=C+C\\ \\ \implies\bf\:C_2=2C$

↗ Equivalent of part C (series) :

$\implies\sf\:\dfrac{1}{C_3}=\dfrac{1}{C_2}+\dfrac{1}{2C}\\ \\ \implies\sf\:\dfrac{1}{C_3}=\dfrac{1}{2C}+\dfrac{1}{2C}=\dfrac{2}{2C}\\ \\ \implies\bf\:C_3=C$

↗ Equivalent of part D (parallel) :

$\implies\sf\:C_4=C_3+C\\ \\ \implies\sf\:C_4=C+C\\ \\ \implies\bf\:C_4=2C$

↗ Equivalent of part E (series) :

$\implies\sf\:\dfrac{1}{C_{eq}}=\dfrac{1}{C_4}+\dfrac{1}{C}\\ \\ \implies\sf\:\dfrac{1}{C_{eq}}=\dfrac{1}{2C}+\dfrac{1}{C}=\dfrac{1+2}{2C}\\ \\ \implies\underline{\boxed{\bf{\pink{C_{eq}=\dfrac{2C}{3}}}}}\:\red{\bigstar}$

⚾ Answer-iv) :

↗ Equivalent of part A (series) :

$\implies\sf\:\dfrac{1}{C_1}=\dfrac{1}{4C}+\dfrac{1}{2C}+\dfrac{1}{C}\\ \\ \implies\sf\:\dfrac{1}{C_1}=\dfrac{1+2+4}{4C}\\ \\ \implies\bf\:C_1=\dfrac{4C}{7}$

↗ Equivalent of part B (series) :

$\implies\sf\:\dfrac{1}{C_2}=\dfrac{1}{2C}+\dfrac{1}{3C}+\dfrac{1}{C}\\ \\ \implies\sf\:\dfrac{1}{C_2}=\dfrac{3+2+6}{6C}\\ \\ \implies\bf\:C_2=\dfrac{6C}{11}$

↗ Equivalent of part C (series) :

$\implies\sf\:\dfrac{1}{C_3}=\dfrac{1}{C}+\dfrac{1}{2C}+\dfrac{1}{C}\\ \\ \implies\sf\:\dfrac{1}{C_3}=\dfrac{2+1+2}{2C}\\ \\ \implies\bf\:C_3=\dfrac{2C}{5}$

↗ Net equivalent (parallel) :

$\implies\sf\:C_{eq}=C_1+C_2+C_3\\ \\ \implies\sf\:C_{eq}=\dfrac{4C}{7}+\dfrac{6C}{11}+\dfrac{2C}{5}\\ \\ \implies\sf\:C_{eq}=\dfrac{220C+210C+154C}{385}\\ \\ \implies\sf\:C_{eq}=\dfrac{584C}{385}\\ \\ \implies\underline{\boxed{\bf{\orange{C_{eq}\approx \dfrac{3C}{2}}}}}\:\green{\bigstar}$