Electromotive Force (EMF):
EMF is defined as the maximum potential difference across a cell's terminals when no current is drawn. It signifies the total energy per unit charge supplied by the cell.
Terminal Voltage:
Terminal voltage is the potential difference across a cell's terminals when it is supplying current. Due to internal resistance \( r \), a voltage drop occurs within the cell. Therefore:
\[ \text{Terminal voltage} = \text{EMF} - Ir \]
Relationship:
\[ \text{EMF} \geq \text{Terminal voltage} \quad (\text{Equality holds when } I = 0) \]
Given: Two cells with EMFs \( E_1 \) and \( E_2 \), and internal resistances \( r_1 \) and \( r_2 \) respectively, connected in parallel.
Objective: To derive the expressions for the equivalent EMF \( E \) and equivalent internal resistance \( r \).
Derivation:
For parallel connections, the terminal voltages of the cells are equal. Let:
\[ E_1 - I_1 r_1 = E_2 - I_2 r_2 = V \]
Let the total current be \( I = I_1 + I_2 \). For the equivalent cell, the relationship is:
\[ V = E - Ir \]
From the current equations:
\[ I_1 = \frac{E_1 - V}{r_1}, \quad I_2 = \frac{E_2 - V}{r_2} \]
The total current is the sum of individual currents:
\[ I = \frac{E_1 - V}{r_1} + \frac{E_2 - V}{r_2} \Rightarrow I = \frac{E_1}{r_1} + \frac{E_2}{r_2} - V\left(\frac{1}{r_1} + \frac{1}{r_2} \right) \]
Substitute this expression for \( I \) into the equivalent cell equation \( V = E - Ir \):
\[ V = E - r\left( \frac{E_1}{r_1} + \frac{E_2}{r_2} - V\left( \frac{1}{r_1} + \frac{1}{r_2} \right) \right) \]
Solving for \( E \) and \( r \) yields:
Equivalent EMF:
\[ E = \frac{\frac{E_1}{r_1} + \frac{E_2}{r_2}}{\frac{1}{r_1} + \frac{1}{r_2}} \]
Equivalent Internal Resistance:
\[ \frac{1}{r} = \frac{1}{r_1} + \frac{1}{r_2} \]