Question

A wire of resistance R is bent to form a square ABCD as shown in the figure. The effective resistance between E and C is:
(E is midpoint of arms CD)

\[A.\,R\]
\[B.\,\dfrac{1}{16}R\]
\[C.\,\dfrac{7}{64}R\]
\[D.\,\dfrac{3}{4}R\]

Answer 1

Hint: Firstly, find the value of the resistance of the sides of the square formed by the wire. The equivalent resistance will be the sum of the individual resistance values of the sides of the square. Then, the effective resistance between E and C can be calculated by using the formula for calculating the parallel resistance value.
Formula used:
\[\begin{align}
  & {{R}_{S}}={{R}_{1}}+{{R}_{2}}+... \\
 & \dfrac{1}{{{R}_{P}}}=\dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}}+... \\
\end{align}\]

Complete answer:
Firstly, we will find the value of the resistance of the sides of the square formed by the wire having the resistance value R.
The resistance of the sides of the square is in series. Thus, we will make use of the series resistance formula.
\[{{R}_{EDABC}}=\dfrac{R}{4}+\dfrac{R}{4}+\dfrac{R}{4}+\dfrac{R}{8}=\dfrac{7R}{8}\]
The value of the resistance of the part EC should be calculated separately, so, we have,
\[{{R}_{EC}}=\dfrac{R}{8}\]
The effective resistance between E and C of the square is calculated as the parallel series. Thus, we will make use of the parallel resistance formula.
\[\begin{align}
  & {{R}_{eff}}=\dfrac{\dfrac{R}{8}\times \dfrac{7R}{8}}{\dfrac{R}{8}+\dfrac{7R}{8}} \\
 & \Rightarrow {{R}_{eff}}=\dfrac{7R}{64} \\
\end{align}\]
Therefore, the value of the effective resistance between point E and C on the square formed by a wire of resistance R is \[\dfrac{7R}{64}\].
As the value of the effective resistance between the points E and C is obtained to be equal to \[\dfrac{7R}{64}\].

So, the correct answer is “Option C”.

Note:
In this case, the wire is bent in the form of a square, thus, the whole resistance should be divided by 4 to find the value of the resistance of the sides of a square. If in case, the wire is bent in the form of a rectangle, thus, the half the value of the whole resistance should be divided by 2 to find the value of the resistance of the sides of a rectangle.