Introduction
A potentiometer is one of the most flexible and precise tools that is used to measure electrical values. A potentiometer measures a potential difference without affecting the circuit, as opposed to a conventional voltmeter, which draws a small current through the source. This ability makes it essential in the comparison of electromotive forces (EMFs) of the electrochemical cells, the measurement of the internal resistances, and the determination of small voltage differences with an outstanding sense of precision.
In short, the principle on which the potentiometer is based consists of the fact that the potential drop across a uniform conductor varies in direct proportion to length when a fixed current is used. This fact of elementary character gives the instrument a strong power of electrical experimentation of high precision.
What is a Potentiometer?
A potentiometer is a piece of uniformly resistive wire, often several metres long, with its ends across a given source of voltage. The wire is mounted on a wooden scale to be measured accurately, and copper strips on its terminals enable connection to outside circuitry.
A sliding contact known as a jockey is used to contact different points along the wire so that researchers can find the point at which no current passes through the galvanometer. This is a state of equilibrium, referred to as the null point, on which the principles of measurement of the potentiometer are based.
Principle of Potentiometer
The fundamental principle of the potentiometer can be stated as:
The potential drop across a uniform wire is directly proportional to its length, provided the wire carries a constant current.
If the total length of the wire is L, the total potential difference across it is V, and the potential drop per unit length is ϕ, then the potential difference across a length l is:
ε(l) = ϕl
where,
- ε(l) = potential difference across length l
- ϕ = potential drop per unit length = V/L
This direct proportionality enables precise measurement by simply comparing lengths.
Construction of a Potentiometer
The construction of a potentiometer is straightforward but carefully designed for accuracy. As shown in Fig. 3.28, the key components include:
- Uniform wire: A long wire of constant cross-sectional area and high resistance, typically made of manganin or constantan, is used to ensure minimal temperature effects.
- Rheostat (R): Placed in series with the wire to adjust the current through it.
- Galvanometer (G): A sensitive detector that shows whether current is flowing in the secondary circuit.
- Standard cell: Provides a stable emf for calibration.
- Keys (K₁, K₂): Switches for connecting or disconnecting circuits during experiments.
- Jockey: A sliding metallic contact with a knife-edge, used to tap different points along the potentiometer wire.
When a current flows through the wire, the potential difference between any two points depends only on the length of the wire between them.
Working of a Potentiometer
To understand its operation, consider a potentiometer wire AC with a steady current. Suppose a cell of emf ε is connected in such a way that its emf is balanced by a length l of the wire. At the balance point:
ε = ϕl
At this point, there is no current through the galvanometer, because the potential difference across the wire length exactly equals the emf of the cell.
This null-deflection condition makes the measurement free from errors caused by the internal resistance of the cell.
Applications of the Potentiometer
The real strength of the potentiometer lies in its multiple applications.
1. Comparing EMFs of Two Cells
One of the most common applications is comparing the emfs of two cells.
Experimental Setup (Fig. 3.28a):
- The two cells (with emf ε1 and ε2) are connected one at a time using a two-way key.
- For the first cell, the balance length l1 is measured.
- For the second cell, the balance length l2 is measured.
Derivation:
From the principle,
ε1=ϕl1, ε2=ϕl2
Dividing,
ε1/ε2 = l1/l2
Thus, the ratio of emfs is simply the ratio of the corresponding balance lengths.
2. Measuring Internal Resistance of a Cell
The potentiometer can also measure the internal resistance of a cell.
Experimental Setup (Fig. 3.28b):
- The cell with emf ε is first balanced directly, giving balance length l1.
- Then, the cell is connected across a resistance box R through a key K2. The new balance length is l2.
Derivation:
When external resistance is connected,
Since ε = V + Ir and V = IR,
Rearranging,
This gives the internal resistance of the cell.
3. Measuring Voltage Drop in a Circuit
The potentiometer can also measure small potential differences across components of a circuit. This is achieved by connecting the component across the potentiometer wire and finding the balance length.
Mathematical Derivations
For convenience, let's summarize the key equations:
- Potential difference across length l:
- Comparison of emfs:
- Internal resistance of a cell:
These equations form the mathematical backbone of potentiometer experiments.
Solved Example
Example 3.10 (Fig. 3.29):
A resistance of R Ω draws current from a potentiometer. The potentiometer has a total resistance R0 Ω. A voltage V is supplied. Derive an expression for the voltage across R when the sliding contact is at the midpoint.
Solution:
- At the midpoint, half of the potentiometer resistance (R0/2) lies between points A and B.
The equivalent resistance between A and B is:
- Total resistance of AC is:
- Current through potentiometer:
- Voltage across R:
Thus,
Advantages of a Potentiometer
- High accuracy: It does not draw current from the source under measurement.
- Wide applications: Can measure emf, internal resistance, and voltage drops.
- Sensitivity: Very small potential differences can be detected.
Limitations of the Potentiometer
- Requires a stable and constant current source.
- The apparatus is relatively bulky compared to simple voltmeters.
- Accuracy depends heavily on the uniformity of the potentiometer wire.
Conclusion
The potentiometer is one of the basic tools in scientific laboratories, combining simplicity and accuracy. Its use in the comparison of electromotive forces in the determination of internal resistance and voltage drops provides a methodology without the limitations that are in the use of conventional voltmeters.
By explaining the principle on which it is based, and how it is carried out, and by considering the examples of its applications resolved, e.g., those which are shown in Figures 3.28 and 3.29, students and researchers can quite properly estimate its permanent importance in the measurements of electricity.