Meter Bridge, P.O.Box and Potentiometer

Meter Bridge

A meter bridge is an electric device used for measuring an unknown resistance. It works in the principle of the balanced condition of Wheatstone bridge.

The device consists of a meter long wire AC of uniform cross-section and of low temperature coefficient of resistance. It is kept stretched on a wooden board between the stripes and an unknown resistance X to the right gap as shown in the figure. A galvanometer is connected between D and a slider which can slide over the wire AC. B is the point at which slider touches to the wire AC and that point can be varied through the wire. A meter scale fixed on the board along AC is used to measure the balancing length. A cell connected across the bridge wire maintains a current is the circuit. The null point is obtained by sliding the jockey on the wire.

\begin{align*} \text {According to Wheatstone bridge principle,} \\ \frac PQ &= \frac RX \\ \text {or,} \: \frac {\text {Resistance of length AB}}{\text {Resistance of length BC}} &= \frac RX \\ \text {let l be the length of wire between A and B,} \\ \text {and then} \: (100 – l) \: \text {is the length between B and C.} \\ \text {Here,} \\ P &= \frac {\rho l}{A} \\ \text {Since the wire has }\text{uniform cross-section and} \: \rho \: \text { is constant ,} \\ \text {its resistance is proportional to the length.}\\ \text {That is} \: P \propto l, \: \text {and} \: Q \propto (100 – l). \: \text {So,} \\ \frac {l}{(100 – l)} &= \frac RX \\ \text {or,} \: X &= \frac {(200 – l) R}{l} \\ \end{align*}

The value of X is calculated for different values of R and the mean value gives the value of unknown resistance X.

P.O.Box

A P.O. box is a compact form of Wheatstone bridge. It can be used to measure low or high resistance.

The box consists of three arms P, Q and R. the value of resistance in P and Q are 10 W, 100 W and 1000 W and in R, the resistance vary from one ohm to 5000 W. So value of R can be adjusted to any integral value from 1 W to 5000 W. Besides, there is the lowest resistance of 0 W and highest value of infinite ohm in R. Two tap keys K₁ and K₂ connect the battery and the galvanometer internally in the circuits respectively. The unknown resistance x is connected between A and D. at null deflection of the galvanometer, the ratio of the resistances must be

\begin{align*} \frac PQ &= \frac XR \\ \text {or,} \: X &= \frac PQ . R \\ \end{align*}

Specific Resistance of X

The specific resistance of X can be measured using .O. Box if the length l and diameter d of the wire X are measured. Then, the specific resistance, ρ is given by

\begin{align*} \rho &= \frac {XA}{l} = \frac {\pi d^2X}{4l} \\ \end{align*}

Potentiometer

A potentiometer is an electric device used to measure the emf and internal resistance of a cell, to compare emf of two cells and a potential difference between two points in an electric circuit. It can also be employed to measure the current and resistance in a circuit accurately.

It consists of a uniform wire AB of manganin or constantan of length usually 10 m, kept stretched between copper stripes fixed on a wooden board by the side of a metre scale. The wire is divided into ten segments each of 1 m length. These segments are joined in series through metal strips between points A and B. A steady current is maintained in the wire AB by a constant source of emf E_o, called driver cell, that is connected between A and B through a rheostat. A jockey is slided over the potentiometer wire which makes contact with the wire and cell.

\begin{align*} \text {let I be the current passing through the potentiometer wire AB.} \\ \text {In a segment AC of the wire, let V be the potential difference across it. } \\ \text {Then, } \\ V &= IR \dots (i) \\ \text {where R is the resistance of this segment.}\\ \text {If l is the length of this segment, its resistance is} \\ R &= \rho \frac {1}{A_c} \dots (ii) \\ \text {where} \: \rho \: \text {to the resistivity of the material of the wire and} \: A_c \: \text {its cross section area.} \\ \text {From}\: (i) \: \text {and} \: (ii), \: \text {we have} \\ v &= \frac {I\rho }{A} l = kl \\ \text {Since} \: \rho, A_c \: \text {and I are all constant,} \\ \text {then} \\ \: K = \frac {\rho I}{A} \: \text { is a proportionality constant and we can write} \\ V \propto l \\ \end{align*}

The potential difference across any portion of the potential of the potentiometer wire is directly proportional to the length of that portion provided the current is uniform.

reference

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.