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Question: The circuit shown in Fig. P26.74, called a Wheatstone Bridge, is used to determine the value of an unknown resistor X by comparison with three resistors M, N, and P whose resistances can be varied. For each setting, the resistance of each resistor is precisely known. With switches S₁ and S₂ closed, these resistors are varied until the current in the galvanometer G is zero; the bridge is then said to be balanced. (a) Show that under this condition the unknown resistance is given by X = MP/N. (This method permits very high precision in comparing resistors.) (b) If galvanometer G shows zero deflection when M = 850.0 鈩�, N = 15.00 鈩�, and P = 33.48 鈩�, what is the unknown resistance X?

Step by step solution

01

About Wheatstone bridge.

The wheat stone bridge is balanced, which means the current through two resistors n and M is the same.

$I_N=I_M=\frac{蔚}{N+M}$ (1)

Also the current through two resistors P and X is the same

$I_P=I_X=\frac{蔚}{P+X}$ (2)

To create the balance in the circuit, the potential at point b is the same for point c. So, we can find the expression of X by using equations(1) and (2)

$$\begin{gathered} V_b=V_c \\ {I}_{N}N=I_{P}P \\ \left(\frac{蔚}{N+M}\right)N=\left(\frac{蔚}{P+X}\right)P \\ N\left(P+X\right)=P\left(N+M\right) \\ NX=MP \\ X=\frac{MP}{N} \end{gathered}$$

02

Calculation

We can get an expression of the unknown resistance X in part a by

$$\begin{gathered} X=\frac{M脳P}{N} \\ =\frac{850惟脳33.48惟}{15惟} \\ =1897惟 \end{gathered}$$

Hence the value of X is $1897惟$

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