Problem

Find the terminal potential difference of each cell in the circuit below:

Data

• Resistances: $r_1 = 0.1 \, \Omega$, $r_2 = 0.9 \, \Omega$, $R = 8 \, \Omega$
• EMF: $\varepsilon_1 = 24 \, \mathrm{V}$, $\varepsilon_2 = 6 \, \mathrm{V}$

Prerequisite Concept

• Net EMF:
  $\varepsilon_{\text{net}} = \varepsilon_1 - \varepsilon_2$
• Net Resistance:
  $R_{\text{net}} = r_1 + R + r_2$

Solution

1. Total EMF:
   $\varepsilon_{\text{net}} = \varepsilon_1 - \varepsilon_2 = 24 - 6 = 18 \, \mathrm{V}.$

2. Total Resistance:
   $R_{\text{net}} = r_1 + R + r_2 = 0.1 + 8 + 0.9 = 9 \, \Omega.$

3. Current:
   $I = \frac{\varepsilon_{\text{net}}}{R_{\text{net}}} = \frac{18}{9} = 2 \, \mathrm{A}.$

4. Terminal Voltage for Cell 1:
   $V_{t1} = \varepsilon_1 - I r_1 = 24 - (2)(0.1) = 23.8 \, \mathrm{V}.$

5. Terminal Voltage for Cell 2:
   $V_{t2} = \varepsilon_2 + I r_2 = 6 + (2)(0.9) = 7.8 \, \mathrm{V}.$

Answer

• Terminal Voltage for Cell 1: $V_{t1} = 23.8 \, \mathrm{V}$
• Terminal Voltage for Cell 2: $V_{t2} = 7.8 \, \mathrm{V}$

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