# Wheatstone Bridge Working Principle

Wheatstone devised a bridge-arrangement of resistances by which the resistance of a given conductor can be determined. This arrangement is called the ‘Wheatstone bridge‘.

In the arrangement, four resistances are so connected as to form a parallelogram. In one diagonal of this parallelogram is connected a galvanometer and in other diagonal a cell.

Now, if the resistances in the four arms of the parallelogram are so adjusted that on sending current in the bride by the cell, there is no deflection in the galvanometer then the bride is said to be ‘balanced’.

In this condition, the ratio of the resistances of any two adjacent arms of the bridge is equal to the ratio of the resistances of the remaining two adjacent arms.

## Wheatstone Bridge Derivation

Applying Kirchhoff’s second law for the closed loop ABDA, we have

-i_{1}P + i_{2}R = 0 ……..(i)

Similarly, for the closed loop BCDB, we have

-i_{1}Q + i_{2}S = 0 ………(ii)

Dividing equation (i) by equation (ii), we have

**P/Q = R/S**

It is clear from this formula that if the ratio of the resistances **P** and **Q** the resistance **R** are known then the unknown resistance **S** can be calculated. This is why the arms P and Q are called ‘ratio arms’, arm **AD** ‘known arm’ and arm **CD** ‘unknown arm’.

When the bridge is balanced then on interchanging the positions of the galvanometer and the cell there is no effect on the balance of the bridge. Hence, the arms **RD** and **AC** are called ‘conjugate arms’ of the bridge.

In the balanced state, no current flows in the galvanometer arm, therefore** the result will be independent of galvanometer resistance**.

The sensitivity of the bridge depends upon the values of the resistances. **The bridge is maximum sensitive when all the resistances are of the same order**.

# Meter Bridge Working Principle

Meter-bridge is a sensitive device based on the *principle of Wheatstone’s bridge*, for the determination of the resistance of a conductor (wire).

Meter-bridge is shown in Figure. **AC** is a 1-meter wire of manganin or constantan which is fixed along a scale on a wooden base. The area of cross-section of the wire is the same at all places. The ends **A** and **C** of the wire are joined to two **L**-shaped copper strips carrying binding-screws as shown.

In between these strips, leaving a gap on either side, there is a third copper strip having three binding screws. The middle screw **D** is connected to a sliding jockey **B** through a shunted-galvanometer **G**. The knob of the jockey can be made to touch at any point on the wire.

## Determination of Resistance by Meter Bridge

The wire whose resistance (S) is to be determined is connected across the gap between the points **C **and **D** and a resistance box across the gap between the points **A** and **D**.

In the experiment, when the sliding jockey touches the wire **AC** at any point **B**, then the wire is divided into two parts. These two parts **AB** and **BC** act as resistances **P** and **Q** of the Wheatstone’s bridge.

First of all, a resistance **R** is taken in the resistance box and the key **K** is closed. Now the jockey is slided along the wire and a point is determined such that, on pressing the jockey on the wire at that point, there is no deflection in the galvanometer **G**.

In this position, the points **B** and **D** are at the same potential. The point **B** is called null-point. The lengths of both the parts **AB** and **BC** of the wire are read on the scale.

Suppose the resistance of the length **AB** of the wire is **P**, and that of the length **BC** is **Q**. Then, by the **principle of Wheatstone bridge**, we have

P/Q = R/S ……(i)

Let the length **AB** be **L** cm. Then the length **BC** will be (100 – L).

Therefore, resistance of **AB**, P = ρL/a

And resistance of **BC**, Q = ρ(100 – L)/a

Where ρ (ohm-cm) is the specific resistance of the material of the wire and a (in cm^{2}) is the area of cross-section of the wire. Thus,

P/Q = L/(100 – L)

Substituting this value of P/Q in equation (i), we get

**L/(100 – L) = R/S**

Here, **R** is the resistance taken in the resistance box and **L** is the length measured. Hence, the value of resistance **S** can be determined from the above formula.

A number of observations are taken for different resistances in the resistance box and for each observation, the value S is calculated.

Finally, the experiment is repeated by interchanging the unknown resistance **S **and the resistance box. The mean of the values of **S** is then obtained.

## Errors and their Removal

1. The resistances of the copper strips fitted at the ends of the wire and that of the solder have not been taken into account. These are called ‘end-resistances’. These are determined by another experiment in terms of the length of the wire and are added in the lengths of the two parts **AB** and **BC** of the wire.

2. If the end of the wire of the meter-bridge is not exactly at the zero-point of the meter scale, or the knife-edge of the sliding jockey (which touches the wire) and its index point (by means of which the scale is read) are not in line, then the lengths of the two parts as and BC read on the scale will be different than the real lengths.

To remove this error, the experiment is repeated after interchanging the unknown resistance **S **and the resistance-box. By doing so, error due to any non-uniformity in the thickness of the bridge-wire is also minimized.

3. The resistance **R** taken in the resistance-box is so chosen that the null-point is nearly in the middle of the wire. Then the percentage error will be least. In addition to it, the effect of end-resistances will also be minimum. There is one more advantage. When the null-point is in the middle, the **sensitivity of the meter bridge** becomes maximum because then all the four resistances (P, Q, R, and S) become nearly equal.

4. The current should not be allowed to flow in the wire for a long time otherwise the wire will become hot and its resistance will be changed.

5. In the initial adjustments, the shunt should be used with the galvanometer. Near the zero-deflection position, the shunt should be removed.

6. The sliding-jokey should not be rubbed on the wire otherwise the thickness of the wire will not remain the same at all places.

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