PX157 - B14a - circuits - kirchhoff's rules

kirchhoff's junction rule

current into element = current out of element:

I_{i n} = I_{o u t}

no charge accumulation

\begin{aligned} I_{1} & = \iint_{A_{1}} \vec{J} \cdot d \vec{S} \\ I_{2} & = \iint_{A_{2}} \vec{J} \cdot d \vec{S} \\ Q & = ∭_{V o l .} ρ d τ \end{aligned}

0 = \frac{d Q}{d t} = \frac{d}{d t} ∭_{V o l .} ρ d τ = \iint_{A_{1}} \vec{J} \cdot d \vec{S} - \iint_{A_{2}} \vec{J} \cdot d \vec{S} = I_{1} - I_{2}

where, $\frac{d Q}{d t}$ is the rate of change of total charge in volume, not rate of charges passing a surface in unit time

taking a limit of decreasing the distance between $A_{1}$ and $A_{2} :$

\begin{aligned} \frac{d Q}{d t} & = I_{t o t} = 0 \\ I_{t o t} = \sum_{i} I_{i} & = \sum_{i, i n} I_{i} + \sum_{i, o u t} I_{i} = 0 \end{aligned}

conventionally, currents toward a circuit are positive, and away are negative

kirchhoff's loop rule

in a DC circuit, the potential difference of a closed path is zero
from here :

\begin{aligned} V_{a} - V_{b} & = \int_{a}^{b} \vec{E} \cdot d \vec{l} \\ \oint \vec{E} \cdot d \vec{l} = \int_{a}^{a} \vec{E} \cdot d \vec{l} & = V_{a} - V_{a} = 0 \end{aligned}

over a circuit loop, the sum of the potentials from sources balances against potential drops across all other components:

\sum_{i} ϵ_{i} = \sum_{j} V_{j}

eg:
- junction rule:
  - at junction $a :$

I_{1} = I_{2} + I_{3}

- at junction $b :$

I_{2} + I_{3} = I_{4} = I_{1}

- loop rule:
- for loop $1 :$

ϵ_{1} = R_{1} I_{4} + R_{2} I_{2}

- for loop $2 :$

ϵ_{2} - ϵ_{3} = - R_{2} I_{2} + R_{3} I_{3}