Derive a relationship between emf, terminal voltage and internal resistance.

Let's derive the relationship between the EMF (Electromotive Force), terminal voltage, and internal resistance of a battery or source.
Key Terms:
• EMF (E): The electromotive force is the maximum potential difference across the terminals of the battery when no current is flowing.
• Terminal voltage (V): The potential difference across the terminals of the battery when the current is flowing through the circuit.
• Internal resistance (r): The resistance inside the battery that causes a drop in the voltage when current flows.
Derivation:
Consider a simple circuit where a battery with EMF EE and internal resistance rr is connected to an external load resistance RR.
1. Current in the circuit:
The total resistance in the circuit is the sum of the internal resistance rr and the load resistance RR:
Rtotal=R+rR_{\text{total}} = R + r
According to Ohm's Law, the current II flowing through the circuit is given by:
I=ERtotal=ER+rI = \frac{E}{R_{\text{total}}} = \frac{E}{R + r}
2. Voltage drop across internal resistance:
When current flows, there is a voltage drop across the internal resistance rr due to the current II. This voltage drop is:
Vdrop=I⋅r=E⋅rR+rV_{\text{drop}} = I \cdot r = \frac{E \cdot r}{R + r}
3. Terminal voltage:
The terminal voltage VV is the potential difference across the external load resistance RR. It is the EMF minus the voltage drop across the internal resistance:
V=E−Vdrop=E−E⋅rR+rV = E - V_{\text{drop}} = E - \frac{E \cdot r}{R + r}
4. Simplifying the expression for terminal voltage:
Factor out EE from the terms on the right-hand side:
V=E(1−rR+r)V = E \left( 1 - \frac{r}{R + r} \right)
This equation gives the relationship between the EMF EE, the terminal voltage VV, the internal resistance rr, and the external resistance RR.
Final Relation:
V=E(RR+r)V = E \left( \frac{R}{R + r} \right)
This is the desired relationship between the EMF, terminal voltage, and internal resistance.
Conclusion:
• EMF (E) is the maximum voltage the battery can provide when no current is flowing.
• Terminal voltage (V) is the voltage across the battery's terminals when current is flowing, and it is less than the EMF due to the voltage drop across the internal resistance.
• Internal resistance (r) causes the voltage to decrease when current flows, and it can be calculated using the difference between the EMF and the terminal voltage.