Potentiometer

Overview

A potentiometer is a circuit used to measure or compare emf and potential difference accurately using the null method.

Instead of reading voltage directly from a meter, an unknown source is balanced against the potential drop along a uniform wire carrying current.

Main uses:

• compare emf of cells
• measure unknown emf
• measure potential difference accurately
• determine internal resistance of a cell

This topic supports DC Circuits.

Definition

A potentiometer is a null-method circuit that measures or compares emf by balancing it against the potential drop along a uniform wire.

Why It Matters

The potentiometer matters because it:

• measures emf without drawing current from the test cell at balance
• compares emfs accurately
• supports internal-resistance measurements
• makes voltage measurement depend on length rather than meter calibration

Key Representations

Core Principle

A long uniform resistance wire carries a steady current.

Because the wire has:

• uniform material
• constant cross-sectional area
• constant current

the potential drop is proportional to length.

$$V\propto l$$

Hence:

$$V=kl$$

where:

• $k$ = potential gradient (V m${}^{-1}$)
• $l$ = length measured along the wire

Potential Gradient

If a wire of total length $L$ has p.d. $V_{\text{wire}}$ across it:

$$k=\frac{V_{\text{wire}}}{L}$$

So over any length $l$:

$$V=kl$$

This converts voltage measurement into length measurement.

Null Method

An unknown cell is connected through a galvanometer to a sliding contact (jockey) on the wire.

The jockey is moved until galvanometer shows zero deflection.

At balance:

• no current flows in the test branch
• potential difference across galvanometer is zero
• unknown emf equals wire voltage over balancing length

$$E=kl$$

Why Null Method Is Accurate

At balance, no current is drawn from the unknown cell.

Therefore:

• no lost volts inside the cell during measurement
• measured value is close to true emf
• galvanometer only detects balance, not magnitude

This is often more accurate than using a voltmeter.

Basic Circuit Components

A typical potentiometer setup includes:

• driver cell or power supply
• rheostat (to control current)
• long uniform wire
• jockey
• galvanometer
• switch
• test cell

Comparing emfs of Two Cells

If two cells balance at lengths $l_1$ and $l_2$ using the same wire current:

$$E_1=k{l}_1$$ $$E_2=k{l}_2$$

Hence:

$$\frac{E_1}{E_2}=\frac{l_1}{l_2}$$

No need to know $k$ explicitly.

Worked Example 1

Two cells balance at lengths:

• $l_1=60\text{ cm}$
• $l_2=45\text{ cm}$

If $E_2=1.50\text{ V}$, find $E_1$.

$$\frac{E_1}{1.50}=\frac{60}{45}$$ $$E_1=2.00\text{ V}$$

Measuring Unknown emf

If gradient $k$ is known and balance length is $l$:

$$E=kl$$

Worked Example 2

Wire length $1.00\text{ m}$ has p.d. $5.0\text{ V}$ across it.

Balance length of unknown cell:

$$l=0.36\text{ m}$$

Step 1: Gradient

$$k=\frac{5.0}{1.00}=5.0{\text{ V m}}^{-1}$$

Step 2: emf

$$E=kl=5.0(0.36)=1.8\text{ V}$$

Measuring Internal Resistance of a Cell

See also Internal Resistance.

Method

Measure:

• emf $E$ with cell not supplying current
• terminal p.d. $V$ when current flows through an external resistor

Using balance lengths:

$$\frac{E}{V}=\frac{l_1}{l_2}$$

Then use:

$$V=\mathcal{E}-Ir$$

or equivalent circuit relations to determine $r$.

Why No Current in Balance Branch Matters

If no current flows through the test cell branch:

• no energy is wasted there
• galvanometer reads zero
• branch resistance does not affect result significantly

This is the essential advantage of potentiometers.

Practical Reasoning

If Wire Current Increases

Potential gradient $k$ increases.

So the same emf balances at a shorter length.

If Unknown emf Increases

Longer balancing length is required.

Common Exam Question Types

1. Compare Two Cells

Use:

$$\frac{E_1}{E_2}=\frac{l_1}{l_2}$$

2. Find Unknown emf

Use:

$$E=kl$$

3. Determine Internal Resistance

Use open-circuit and loaded balance lengths.

4. Explain Zero Deflection

Balanced potentials produce zero p.d. across galvanometer.

Common Mistakes

1. Thinking No Current Flows Anywhere

Only the test branch has zero current at balance. Current still flows in the main wire.

2. Wrong Ratio Direction

Larger emf gives larger balancing length.

3. Confusing emf with Terminal p.d.

Loaded cell may not give full emf.

4. Ignoring Need for Uniform Wire

Linear relation depends on uniform wire.

Quick Comparison Table

Quantity Increased	Effect
Unknown emf	longer balance length
Wire current	larger gradient
Wire current	shorter balance length for same emf

Links

• DC Circuits
• Potential Divider
• Internal Resistance
• Circuit Fault Finding
• DC Circuits Common Exam Traps

Summary

A potentiometer is a precise null-method device that balances an unknown emf against the voltage drop along a uniform wire.

Key ideas:

• potential drop proportional to length
• zero galvanometer deflection at balance
• no current drawn from test cell
• excellent for emf comparison and internal resistance measurement

This is one of the most elegant DC-circuit techniques in H2 Physics.