What are Kirchoff's Laws?

Kirchhoff's laws are fundamental principles used to determine current and p.d. at specific parts of complicated circuits. 

Kirchoff's First Law

  • The total current entering any point in a circuit must equal to the total current leaving that same point.
  • This is a consequence of the conservation of charge, meaning that the amount of charge entering a point must be the same as the amount of charge exiting.
  • For example, if a 5.0 A current enters a junction and splits into two branches, the sum of the currents in the two branches must be 5.0 A.
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Kirchoff's Second Law

  • The sum of the electromotive forces (e.m.f.s) around any closed loop in a circuit is equal to the sum of the potential differences (p.d.s) around the same loop.
  • This law is a consequence of the conservation of energy.
  • If a charge moves around a loop, it gains energy from e.m.f. sources and loses energy across components with p.d.s, and the total energy gained must equal the total energy lost.
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Using Kirchoff's Laws to solve problems



$$\begin{align*}I_1 + I_2 &= I_3 \quad \text{(Equation 1)}\\\text{Loop 1:}\\\text{Sum of p.d} &= \text{Sum of emf}\\I_3(50) + I_1(10) &= 6\\10I_1 + 30I_3 &= 6 \quad \text{(Equation 2)}\\\text{Loop 2:}\\\text{Sum of p.d} &= \text{Sum of emf}\\I_3(30) &= 2\\I_3 &= \frac{2}{30} = 0.0667 \, \text{A}\\\text{Substitute } I_3 \text{ into Equation 2:}\\10I_1 + 30(0.0667) &= 6\\10I_1 + 2.00 &= 6\\10I_1 &= 4.00\\I_1 &= \frac{4}{10} = 0.4 A\\\text{From Equation 1:}\\I_2 &= I_3 - I_1\\I_2 &= 0.0667 - 0.400\\I_2 &= -0.333 A\\\text{Thus, } I_1 &= 0.4 \, \text{A}, \quad I_2 = -0.333 \, \text{A}, \quad I_3 = 0.0667 \, \text{A} \end{align*}$$

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