Question

Four resistances \[5\,\Omega \], \[10\,\Omega \], \[15\,\Omega \] and an unknown
\[X\,\Omega \] are connected in series so as to form Wheatstone’s network. Determine the
unknown resistance X if the network is balanced with these numerical values of resistance.

Answer 1

Hint:Use the expression for balanced condition in a Wheatstone’s bridge network. This balanced condition gives the relation between the four resistances connected in the Wheatstone’s bridge network. Substitute the values of all the resistances in this balanced condition and determine the value of the unknown resistance connected in the Wheatstone’s network.

Formula used:
The balanced condition for the Wheatstone’s bridge network is given by
\[\dfrac{{{R_1}}}{{{R_2}}} = \dfrac{{{R_3}}}{{{R_4}}}\] …… (1)
Here, \[{R_1}\], \[{R_2}\], \[{R_3}\] and \[{R_4}\] are the resistances connected in the
Wheatstone’s bridge network.

Complete step by step answer:
We have given that the four resistances \[5\,\Omega \], \[10\,\Omega \], \[15\,\Omega \] and an unknown \[X\,\Omega \] are connected in series to form a Wheatstone’s bridge network. We have asked to determine the value of the unknown resistance X.

Let us consider the unknown resistance X is placed at the resistance \[{R_3}\] in the Wheatstone’s bridge network.
\[{R_3} = X\,\Omega \]
Let the values of the resistances \[{R_1}\], \[{R_2}\] and \[{R_4}\] be \[5\,\Omega \],
\[10\,\Omega \] and \[15\,\Omega \] respectively.
\[{R_1} = 5\,\Omega \]
\[{R_2} = 10\,\Omega \]
\[{R_4} = 15\,\Omega \]

The Wheatstone’s bridge network formed by these four resistances is in the balanced condition.
We can determine the value of the unknown resistance X using equation (1).
Substitute \[5\,\Omega \] for \[{R_1}\], \[10\,\Omega \] for \[{R_2}\], \[X\,\Omega \] for
\[{R_3}\] and \[15\,\Omega \] for \[{R_4}\] in equation (1).

\[\dfrac{{5\,\Omega }}{{10\,\Omega }} = \dfrac{{X\,\Omega }}{{15\,\Omega }}\]
\[ \Rightarrow \dfrac{1}{2} = \dfrac{X}{{15}}\]
Rearrange the above equation for X.
\[ \Rightarrow X = \dfrac{{1 \times 15}}{2}\]
\[ \Rightarrow X = 7.5\,\Omega \]

Hence, the value of the unknown resistance is \[7.5\,\Omega \].

Note:One can also solve the same question by another method. One can consider the unknown resistance X placed at the position of any of the four resistances used in the Wheatstone’s bridge as they are all connected in series. One may obtain some other value of the unknown resistance by this method. But the resistor of the obtained value placed at the considered position in the Wheatstone’s network will balance the network.