"Terminal p.d. of a cell or battery",

TL;DR

The terminal potential difference (p.d.) is the actual voltage available at a cell's terminals when it's supplying current to an external circuit. It's always less than the cell's electromotive force (e.m.f.) due to the voltage drop across the cell's internal resistance. Understanding this difference is crucial for designing and analyzing real-world electrical circuits.

1. The Mental Model

Imagine a cell as a tiny factory that generates voltage (e.m.f.). Inside this factory, there's a small "tollbooth" that resists the flow of current (internal resistance). The voltage you actually get to use outside the factory (terminal p.d.) is the original generated voltage minus the voltage lost at that internal tollbooth.

2. The Core Material

When we talk about a cell or battery, we often refer to its electromotive force (e.m.f.), which is the total electrical energy provided by the cell per unit charge. Think of e.m.f. as the maximum potential difference the cell could provide when no current is being drawn from it (i.e., in an open circuit).

However, no real cell is perfect. Every cell has some internal resistance, usually denoted by r. This internal resistance is due to the materials the cell is made from and the chemical processes occurring within it. When current flows through the cell, some of the energy generated by the cell is dissipated within the cell itself as heat due to this internal resistance.

What is Terminal Potential Difference (Terminal p.d.)?

Photo by Wolfgang Weiser on Pexels

The terminal potential difference (V) is the actual voltage measured across the terminals of the cell when it's connected to an external circuit and supplying current. Because of the voltage drop across the internal resistance, the terminal p.d. is always less than the cell's e.m.f. when current is flowing out of the cell.

Let's break it down with some simple relationships:

1. Ohm's Law for the entire circuit: If a cell with e.m.f. (E) and internal resistance (r) is connected to an external load resistance (R), the total resistance in the circuit is R + r. The total current (I) flowing through the circuit is given by:
   I = E / (R + r)

2. Voltage drop across internal resistance: As current I flows through the internal resistance r, there's a voltage drop across it, often called the "lost volts". This lost voltage (V_lost) is calculated using Ohm's Law:
   V_lost = I * r

3. Calculating Terminal p.d.: The terminal p.d. (V) is the e.m.f. minus these lost volts. It's the voltage available to the external circuit:
   V = E - I * r

   Alternatively, you can also think of the terminal p.d. as the voltage across the external load resistance (R). So, applying Ohm's Law to the external circuit:
   V = I * R

These two equations for V are equivalent. If you substitute I = E / (R + r) into V = I * R, you'll get V = E * R / (R + r).

How current affects terminal p.d.

Photo by R Bude on Pexels

The greater the current (I) drawn from the cell, the larger the I * r term becomes, and thus the lower the terminal p.d. (V).
- When I = 0 (open circuit), V = E.
- When I > 0, V < E.
- In the extreme case of a short circuit (where R is almost zero), I becomes very large (I = E / r), and V approaches zero.

Here's a diagram to help visualize the flow of potential difference:

graph TD
    A["Cell E.M.F. (E)"] --> B["Internal Resistance (r)"]
    B --> C["Lost Volts (I * r)"]
    A --> D["Terminal p.d. (V)"]
    D --> E["External Load (R)"]
    C --> F["Resulting Terminal p.d."]
    D --"V = E - I * r"--> F
    E --"V = I * R"--> F

The terminal p.d. is what your devices actually "see" and use to function. It's why a battery can feel warm when supplying a lot of current – that's the I^2 * r power being dissipated internally.

3. Worked Example

Let's say we have a dry cell with an e.m.f. (E) of 1.5 V and an internal resistance (r) of 0.2 Ω.

Scenario: This cell is connected to an external lamp that has a resistance (R) of 4.8 Ω.

Task:
1. Calculate the current flowing through the circuit.
2. Calculate the voltage drop across the internal resistance (lost volts).
3. Calculate the terminal potential difference across the lamp.

Solution:

1. Current (I):
   I = E / (R + r)
I = 1.5 V / (4.8 Ω + 0.2 Ω)
I = 1.5 V / 5.0 Ω
I = 0.3 A

2. Voltage drop across internal resistance (lost volts, V_lost):
   V_lost = I * r
V_lost = 0.3 A * 0.2 Ω
V_lost = 0.06 V

3. Terminal Potential Difference (V):
   Using V = E - I * r:
   V = 1.5 V - 0.06 V
V = 1.44 V

   (Self-check using V = I * R):
   V = 0.3 A * 4.8 Ω
V = 1.44 V

Both methods give the same terminal p.d. of 1.44 V. This means while the cell generates 1.5 V, only 1.44 V is actually available to power the lamp, with 0.06 V being "lost" internally.

4. Key Takeaways

• E.m.f. (E) is the maximum potential difference a cell can provide, measured on open circuit (no current flowing).
• Internal resistance (r) is the resistance within the cell itself, causing some energy loss.
• Terminal p.d. (V) is the actual voltage available at the cell's terminals when current is flowing through an external circuit.
• V = E - I * r is the fundamental relationship showing how terminal p.d. is reduced by lost volts.
• The higher the current drawn (I), the greater the lost volts (I * r), and the lower the terminal p.d. (V).
• When no current flows (I = 0), the terminal p.d. equals the e.m.f. (V = E).

Common Mistakes to Avoid:
- Confusing e.m.f. with terminal p.d.; they are only equal in an open circuit.
- Forgetting to include internal resistance in total circuit resistance calculations.
- Assuming a cell's stated voltage (e.g., "1.5 V battery") is its terminal p.d. under load, rather than its e.m.f.
- Incorrectly applying Ohm's Law: remember I = E / (R_total) for the entire circuit, and V = I * R_external for the external load, or V_lost = I * r for internal drop.

5. Now Try It

A rechargeable battery has an e.m.f. of 6.0 V and an internal resistance of 0.5 Ω. It's connected to a toy car motor with a resistance of 5.5 Ω.

1. Calculate the total resistance of the circuit.
2. Determine the current flowing through the motor.
3. Calculate the terminal potential difference across the motor.

What success looks like: You'll have three clear numerical answers, with units, showing the step-by-step calculations for each. The terminal p.d. should be less than the e.m.f. of 6.0 V.