Question

Twelve capacitors each having a capacitance $C$, are connected to form a cube. Find the equivalent capacitance of the cube along the face diagonal?
[Take $C=9\mu \text{F}$]

• A. $\frac{4}{3}\mu \text{F}$
• B. $\frac{12}{7}\mu \text{F}$
• C. $12\mu \text{F}$
• D. $\frac{5}{6}\mu \text{F}$

Answer 1

The correct option is C $12\mu \text{F}$


Let us consider the face $\text{(1265)}$ to find the effective capcitance of the given network, about its diagonal.

To solve this, let us consider any two symmetrical path for the chosen diagonal (i.e) $\left(126\right)$ and $\left(156\right)$.

About this path we can redraw the given , $3D$ network into $2D$ network as follows.


As we know, about the line of symmetry all points on the line drawn perpendicular to the line of symmetry are equipotential (i.e) points $5,8,3$ and $2$ are of same potential, so we can ignore capcitors connected between those points.


Further we can redraw circuit as


Therefore, equivalent capacitance across the face diagonal,

$C_{Eqv}=\frac{C}{2}+\frac{C}{3}+\frac{C}{2}=\frac{4}{3}C$

As, $C=9\mu \text{F}$,

$\Rightarrow C_{Eqv}=\frac{4}{3}\times 9$

$\therefore C_{Eqv}=12\mu \text{F}$

Hence, option (c) is correct answer.

Why this question ? About face diagonal talking any two symmetrical paths, we can redraw the given 3D network into 2D network.