Post Office Box (Wheatstone Bridge)

A Post Office Box is a precision instrument based on the Wheatstone Bridge, used for accurately measuring unknown electrical resistance. It contains fixed ratio arms (P and Q), a variable resistance arm (R), and plug-in blocks that allow quick selection of resistance values. Widely used in physics and electrical laboratories, the Post Office Box provides high accuracy, easy operation, and reliable resistance measurements through the bridge-balance method.

1.0Definition of Post Office Box

• The Post Office Box is a practical device used for precisely measuring an unknown electrical resistance (X) by applying the Wheatstone Bridge principle.

2.0Principle of Post Office Box

Post Office Box also works on the principle of Wheatstone’s bridge.

The core principle relies on the balance condition of the Wheatstone bridge, shown in the simple diagram

$\frac{P}{Q}=\frac{R}{X}\text{ or }X=\frac{Q}{P}R$

When the bridge is balanced, the galvanometer (G) shows zero deflection, meaning there is no current flowing between points B and D.

3.0Components of Post Office Box

The Post Office Box consists of three key sets of resistance arms:

1. Ratio Arms(P and Q):These are arms AB and BC. They contain plug resistances that allow setting the ratio $\frac{P}{Q}$

• Available Resistance: Typically $(10\Omega ,100\Omega \text{ or }1000\Omega )$
• Function: Used to set a specific ratio (e.g. $1,10,100,\frac{1}{10},\frac{1}{100}$) for the calculation.

2. Resistance Arm(R): This is arm AD. It contains a wide range of plug resistances (often from $1\Omega$ upto $5000\Omega$) in small increments.

• Function: This arm is adjusted to achieve the balance condition. The value of R at balance is read directly from the plugs removed.

3. Unknown Resistance Arm (X): The unknown resistance X is connected across the gap CD.

4.0Procedure For Determining Unknown Resistance

The measurement is a multi-step process involving successive approximations to achieve high accuracy:

Step-1 : Setting the Ratio $\left(\frac{Q}{P}\right)$

• Start by setting a simple ratio,usually $\frac{Q}{P}=1(e\cdot g\cdot P=10\Omega \text{ and }Q=10\Omega )$

Step-2 : Achieving Initial Balance 

• Adjust R: Systematically change the resistance R (by removing plugs in arm AD) until the galvanometer shows zero deflection.
• Balancing Check: To protect the galvanometer, always press the battery key $\left(K_1\right)$ first, and then the galvanometer key $\left(K_2\right)$ .
• Finding the Range:If at $R=4\Omega$, the deflection is towards the left, and at $R=5\Omega$, ,the deflection is towards the right ,the true value of R for balance lies between $4\Omega$ and $5\Omega$.
• Initial Calculation:Using the ratio $\frac{Q}{P}=1$,the unknown resistance X is estimated to be between 

$X=\frac{Q}{P}R\Rightarrow X=1\times R\Rightarrow X\in (4\Omega ,5\Omega )$

Step-3 : Getting Closer to the True Value

• Change the Ratio: To improve precision, select a new ratio that allows for reading R in smaller increments, typically $\frac{Q}{P}=\frac{1}{10}$ (e.g. $P=100\Omega$ and $Q=10\Omega$)
• Refine R: Adjust R in $1\Omega$ interval near the range found in step 2.
• Suppose at $R=42\Omega$, the deflection is left and at $R=43\Omega$, the deflection is right.
• The true balance value of R is between $42\Omega$, and $43\Omega$.

Step-4 : The unknown resistance X is now calculated as:

$X=\frac{Q}{P}R\Rightarrow X\in \frac{10}{100}(42\Omega ,43\Omega )\Rightarrow X\in (4.2\Omega ,4.3\Omega )$

Step-5 : Maximizing Precision Maximum Ratio

To achieve the highest possible accuracy,set the ratio to $\frac{Q}{P}=\frac{1}{100}$ (e.g. $P=1000\Omega$ and $Q=10\Omega$)

• Further Refinement: Adjust R using $1\Omega$ intervals and repeat the deflection check.This will narrow the range of R to a $1\Omega$ interval $(\text{ e.g. }R\in (425\Omega ,426\Omega ))$
• Final Calculation:

$X=\frac{Q}{P}R\Rightarrow X\in \frac{10}{1000}(425\Omega ,426\Omega )$

$\Rightarrow X\in (4.25\Omega ,4.26\Omega )$

By using different ratios $\left(\frac{1}{1},\frac{1}{10},\frac{1}{100}\right)$. The post office box allow the user to find the value of X with increasing Precision.

5.0Common Sources of Errors And Elimination of Errors

Common Sources of Errors

1. Poor contact of resistance plugs: Causes fluctuating resistance.
2. Joule heating: Current for a long time changes R value.
3. Loose or oxidized connecting wires: Adds unwanted resistance.
4. Galvanometer sensitivity not optimal: Hard to detect null point.

6.0 Elimination of Errors

1. Insert plugs firmly and cleanly
2. Use low current (press keys briefly)
3. Use short and thick copper connecting wires
4. Adjust galvanometer for maximum sensitivity

Illustration-1: While finding the null point in a Post Office Box, why must the battery key (K₁)be pressed before the galvanometer key (K₂)?

Solution: If K₂ is pressed before K₁, the sudden current surge when K₁ is closed induces a large back e.m.f. in the galvanometer (due to self-induction), which may damage it—so K₁ is always pressed first to avoid this.