Current Electricity 1 Question 7
7. A wire of resistance $R$ is bent to form a square $A B C D$ as shown in the figure. The effective resistance between $E$ and $C$ is $[E$ is mid-point of arm $C D]$
(2019 Main, 9 April I)
(a) $\frac{7}{64} R$
(b) $\frac{3}{4} R$
(c) $R$
(d) $\frac{1}{16} R$
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Solution:
- Let the length of each side of square $A B C D$ is $a$.
$\therefore$ Resistance per unit length of each side $=\frac{R}{4 a}$
Now, $\quad R _1=\frac{R}{4 a}[E C]=\frac{R}{4 a} \times \frac{a}{2}=\frac{R}{8}$
Similarly, $\quad R _2=\frac{R}{4 a}[E D A B C]=\frac{R}{4 a} \times \frac{7 a}{2}=\frac{7 R}{8}$
Now, effective resistance between $E$ and $C$ is the equivalent resistance of $R _1$ and $R _2$ that are connected in parallel as shown below.
$$ \begin{aligned} R _{E C}=\frac{R _1 R _2}{R _1+R _2} & =\frac{(R / 8) \times(7 R / 8)}{(R / 8)+(7 R / 8)} \\ & =\frac{7 R^{2}}{64} \times \frac{8}{8 R}=\frac{7}{64} R \end{aligned} $$
