Q.A potentiometer wire has a length of 4 m and a resistance of $20\ \Omega$. It is connected in series with a resistance of $2980\ \Omega$ and a cell of emf 4 V. Calculate the potential gradient along the wire.

Potential Gradient: The "Slope" of Voltage Along a Wire

Imagine you're walking down a hill. The steeper the hill, the faster your potential energy drops with every step you take. Now replace "height" with "electric potential" and "walking along the hill" with "moving along a wire." That's the core idea of a potential gradient.

The Intuition

In a potentiometer, we have a long uniform wire with a battery connected across its ends. The battery creates a potential difference $V$ between the two ends of the wire. As you move from one end to the other, the potential doesn't jump — it falls smoothly and steadily along the length.

If the wire is $L$ metres long and the total potential drop is $V$ volts, then every metre of wire you travel, the potential drops by a fixed amount. That fixed drop per metre is the potential gradient.

The Precise Definition

The potential gradient $k$ is defined as the fall of potential per unit length of the potentiometer wire.

where:

• $V$ is the total potential difference across the ends of the wire (in volts)
• $L$ is the total length of the wire (in metres)
• $k$ is the potential gradient (in volts per metre, V/m)

Why It Matters: Sensitivity

The potential gradient directly determines how sensitive your potentiometer is. A smaller $k$ means the potential changes very slowly along the wire — so a small unknown voltage will produce a measurable balancing length. A larger $k$ means the potential drops quickly, and you might not be able to measure small voltages accurately.

You can reduce $k$ by:

• Decreasing the voltage $V$ across the wire (using a smaller driving cell or a rheostat)
• Increasing the length $L$ of the wire

The Formula in Action

Suppose a 10 m long potentiometer wire has a 2 V battery across it. Then:

$k = \frac{2 \text{ V}}{10 \text{ m}} = 0.2 \text{ V/m}$ …