Chapter 26 Problem 71

Problem

The Wheatstone bridge

A Wheatstone bridge is a type of “bridge circuit” used to make measurements of resistance. The unknown resistance to be measured, $R_{x}$ , is placed in the circuit with accurately known resistances $R_{1}$ ,$R_{2}$ , and $R_{3}$. One of these, $R_{3}$, is a variable resistor which is adjusted so that when the switch is closed momentarily, the ammeter A shows a zero current flow.

(a) Determine $R_{x}$ in terms of $R_{1}$ ,$R_{2}$ , and $R_{3}$.

(b) If a Wheatstone bridge is “balanced” when $R_{1}=630\Omega$, $R_{2}=972\Omega$, and $R_{3}=78.6\Omega$, what is the value of the unknown resistance?

Solution

(a)

No current means the potential difference between B and D is zero. Thus

$V_{\textrm {BA}}=<math>V_{\textrm {DA}}$ $\rightarrow I_{3}R_{3}=I_{1}R_{1}$ $\rightarrow {\frac {R_{3}}{R_{1}}}={\frac {I_{1}}{I_{3}}}$

$V_{\textrm {CB}}=<math>V_{\textrm {CD}}$ $\rightarrow I_{3}R_{x}=I_{1}R_{2}$ $\rightarrow R_{x}=R_{2}{\frac {I_{1}}{I_{3}}}={\frac {R_{2}R_{3}}{R_{1}}}$

(b)

$R_{x}={\frac {R_{2}R_{3}}{R_{1}}}={\frac {972\times 78.6}{639}}\Omega =121\Omega$