Question

Question 3 tur mm Figure 2 Figure 2 depicts a circuit diagram featuring a source of EMF and four resistors with resistances as marked. Use Kirchoff's rules to work out the currents i, iz and is (which are defined in the figure) in terms of the EMF ε and the resistances. [18 marks]

Answer 1

Using KVL in loop ABHIA

i₁ R₁ - E = 0

i₁ = E/ R₁

Using KVL in loop ABCDEFGHIA

i₃ R₃ + (i₂ + i₃) R₄ = E

i₃ (R₃ + R₄) + i₂ R₄ = E                                              eq-1

Using KVL in loop BCFGHB

i₂ R₂ + (i₂ + i₃) R₄ - i₁ R₁ = 0

i₂ R₂ + (i₂ + i₃) R₄ = E

i₂ (R₂ + R₄) + i₃ R₄ = E                                            eq-2

Using KVL in loop CDEFC

i₃ R₃ - i₂ R₂ = 0

i₃ = i₂ R₂ /R₃                                                         eq-3

Using eq-2 and eq-3

i₂ (R₂ + R₄) + ( i₂ R₂ /R₃  )R₄ = E  

i₂ (R₂ R₃ + R₄ R₃ + R₂ R₄) = E R₃

i₂ = E R₃ /(R₂ R₃ + R₄ R₃ + R₂ R₄)

using eq-3

i₃ = i₂ R₂ /R₃

i₃ = (E R₃ /(R₂ R₃ + R₄ R₃ + R₂ R₄)) (R₂ /R₃)

i₃ = E R₂ /(R₂ R₃ + R₄ R₃ + R₂ R₄)