The circuit shown in Fig. P 26.81, called a Wheatstone bridge, is used to determine the value of

The circuit shown in Fig. P 26.81, called a Wheatstone...

The circuit shown in Fig. $\mathrm{P} 26.81$, 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 $\mathrm{K}_{1}$ and $\mathrm{K}_{2}$ closed, these resistors are varied until the current in the galvanometer $\mathrm{G}$ is zero; the bridge is then said to be balanced. (a) Show that under this condition the unknown resistance is given by $X=M P / N$. (This method permits very high precision in comparing resistors.) (b) If the galvanometer G shows zero deflection when $M=850.0 \Omega, N=15.00 \Omega,$ and $P=33.48 \Omega,$ what is the unknown resistance $X ?$

The circuit shown in Fig. $\mathrm{P} 26.81$, 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 $\mathrm{K}_{1}$ and $\mathrm{K}_{2}$ closed, these resistors are varied until the current in the galvanometer $\mathrm{G}$ is zero; the bridge is then said to be balanced.
(a) Show that under this condition the unknown resistance is given by $X=M P / N$. (This method permits very high precision in comparing resistors.)
(b) If the galvanometer G shows zero deflection when $M=850.0 \Omega, N=15.00 \Omega,$ and $P=33.48 \Omega,$ what is the
unknown resistance $X ?$

Key Concepts

Wheatstone Bridge

A Wheatstone bridge is a circuit configuration that consists of four resistors arranged in a diamond-like formation. It is primarily used to compare an unknown resistor with known resistors with high accuracy by adjusting the conditions until the bridge is balanced, meaning there is no potential difference between certain nodes of the circuit.

Bridge Balance Condition

The bridge balance condition occurs when the ratio of resistances in one branch of the circuit equals the ratio in the other branch, leading to zero current through the detector (such as a galvanometer). This condition is key to deriving relationships between the resistors, from which the unknown resistance can be expressed in terms of the known resistors.

Kirchhoff's Circuit Laws

Kirchhoff's circuit laws, which include the current law (KCL) and the voltage law (KVL), provide a systematic method for analyzing the distribution of currents and voltages in a circuit. In the context of a Wheatstone bridge, Kirchhoff's laws are used to set up equations for the loops within the circuit, ultimately leading to the derivation of the balance condition.

Null Measurement Technique

The null measurement technique is an approach in measurement where adjustments are made until a measurement device shows zero, indicating that the desired balance or cancellation condition has been achieved. This technique reduces errors related to the measuring instrument's sensitivity and allows for very precise determinations of quantities such as resistance.

Transcript

00:01 Now we are given this circuit which we call a wheat stone bridge in which we know the resistors pn and m very accurately, we know their resistance and they can be varied, whereas x is our unknown, which we want to find.

00:24 And to do that, we need to set the three other resistors in such a way that there is no current flowing through the gubernometer.

00:36 So current is zero.

00:39 There's no current flowing through the gubernometer.

00:41 And so what this means is that only current flows through right over here down or through pnx.

00:54 So either through nnm or pnx.

00:57 There's no crossing in between.

01:01 And this is our condition where the bridge is balanced and it will tell us what is the resistance of x so how and y so first off we need to know that this condition occurs when the potential at this point let us call this va is equal to the potential at this point and what this tells us is that the potential drop across n and the potential drop across p must be the same and that goes for m and x as well all right and to find what is the potential drop you need to find what is the current flowing through and we know that the current is dependent on the emf the battery and the total resistance of the path that you choose so let's say it chooses the left the total resistance is n plus m.

02:08 And so the current i equals to v over r, so that's the emf over the total resistance.

02:17 It is the current flowing through the left side of the path that i'll just call this current i nm.

02:31 On the other hand, for the current passing through p and x, you'll be epsilon over p plus n x.

02:42 Now we can find what is the potential drop.

02:46 Now the potential drop across n will be 3 equals to ir so we take this current multiplied by the resistor this will be epsilon times n for n plus m...

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