## Wheat Stone Bridge Principle:

It states that if four resistances P, Q, R and S are arranged to form a bridge as shown, with a cell E and Key K_{1} connected between the points A and C, and a galvanometer G and Key K_{2} between the points B and D, then on closing K_{1} first and K_{2} later on if galvanometer shows no deflection, then the bridge is balanced.

In that case, P/Q = R/S |

**Proof:** Press the Key K_{1} first and then K_{2}, current I flow through the circuit.

**At point A:** I_{1} current flows through resistance P and (I – I_{1}) current flows through R.

**At point B:** I_{g} current flows through galvanometer G and (I_{1} – I_{g}) current flows through Q.

**At point D:** (I – I_{1}) is coming from R and I_{g} current is coming from G. Hence total current at D is (I – I_{1} + I_{g}) which flows through S as shown.

**At point C:** (I_{1} – I_{g}) current is coming from Q and (I – I_{1} + I_{g}) current is coming from S. Hence total current at C is I which flow through the circuit as shown.

Apply Kirchoff’s loop law to loop ABDA,

I_{1}P + I_{g}G – (I – I_{1}) R = 0 |

The value of R is adjusted such that the galvanometer shows no deflection i.e. I_{g} = 0.

⇒ I_{1}P – (I – I_{1}) R = 0⇒ I _{1}P = (I – I_{1}) R ………………(I) |

Apply Kirchoff’s loop law to loop BCDB,

(I_{1} – I_{g}) Q – (I – I_{1} + I_{g}) S – I_{g} G = 0 |

The value of R is adjusted such that the galvanometer shows no deflection i.e. I_{g} = 0.

(I_{1} – 0) Q – (I – I_{1} + 0) S – 0 = 0⇒ I _{1}Q – (I – I_{1}) S = 0⇒ I _{1}Q = (I – I_{1}) S ……………..(II) |

Divide (I) by (II),

I_{1}P / I_{1}Q = (I – I_{1}) R / (I – I_{1}) S ⇒ P/Q = R/S |