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4.2: Conservation of Charge

  • Page ID
    81489
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    Using the accounting framework, we can now develop the following statement for positive charge and negative charge:

    \[ \left[ \begin{array}{c} \text{ Rate of} \\ \text{ Accumulation } \\ \text{ of } +/- \text{charge} \\ \text{ inside the system } \\ \text{ at time } t \end{array} \right] = \underbrace{ \left[ \begin{array}{c} \text { Transport Rate } \\ \text { of } +/- \text { charge } \\ \text { into the system } \\ \text{ at time } t \end{array}\right] - \left[\begin{array}{c} \text { Transport Rate } \\ \text { of } +/- \text { charge } \\ \text { out of the system } \\ \text{ at time } t \end{array} \right] }_{\text{Net Transport Rate Into The System}} + \underbrace{ \left[ \begin{array}{c} \text { Generation Rate } \\ \text { of } +/- \text { charge } \\ \text { inside the system } \\ \text{ at time } t \end{array}\right] - \left[\begin{array}{c} \text { Consumption Rate } \\ \text { of } +/- \text { charge } \\ \text { inside the system } \\ \text{ at time } t \end{array}\right] }_{\text{Net Generation Rate Inside the System}} \nonumber \]

    In symbols, the rate-form of the accounting equation for positive charge can be written as

    \[\frac{d}{d t} q_{ \ sys}^{+} = \sum_{in} \dot{q}_{i}^{+} - \sum_{out} \dot{q}_{e}^{+} \quad + \quad \dot{q}_{gen}^{+} - \dot{q}_{cons}^{+} \nonumber \]

    In symbols, the rate-form of the accounting equation for negative charge can be written as

    \[\frac{d}{d t} q_{ \ sys}^{-} = \sum_{in} \dot{q}_{i}^{-} - \sum_{out} \dot{q}_{e}^{-} \quad + \quad \dot{q}_{gen}^{-} - \dot{q}_{cons}^{-} \nonumber \]

    If we subtract Equation \(\PageIndex{3}\) from Equation \(\PageIndex{2}\) and apply our definition of net charge, we obtain the following results:

    \[ \frac{d}{dt} \left( q_{\ sys}^{+} - q_{\ sys}^{-} \right) = \sum_{in} \left( \dot{q}_i^{+} - \dot{q}_i^{-} \right) - \sum_{out} \left( \dot{q}_e^{+} - \dot{q}_e^{-} \right) \quad + \quad \left( \dot{q}_{gen}^{+} - \dot{q}_{gen}^{-} \right) - \left( \dot{q}_{cons}^{+} - \dot{q}_{cons}^{-} \right) \nonumber \] \[\frac{d}{dt} q_{sys} = \underbrace{ \sum_{in} \dot{q}_i - \sum_{out} \dot{q}_e }_{\text{Transport across boundaries}} \quad + \quad \underbrace{ \dot{q}_{gen} - \dot{q}_{cons} }_{\begin{array}{c} \text{Generation/Consumption} \\ \text{inside the system} \end{array} } \nonumber \]

    But we know that net charge is conserved so the generation and consumption terms on the right-hand side of Equation \(\PageIndex{5}\) must be zero. Thus Equation \(\PageIndex{5}\) can be written as the rate-form of the conservation of net charge equation:

    \[\frac{d}{d t} q_{sys}=\sum_{in} \dot{q}_{i}-\sum_{out} \dot{q}_{e} \nonumber \]

    This is the primary equation for solving problems involving charge. We will often refer to "charge" without specifically saying "net charge". Unless we explicitly refer to positive or negative charge, the term charge should be interpreted as meaning net charge. We could also write Eq. \(\PageIndex{6}\) using the long established current notation as

    \[\frac{d}{d t} q_{sys}=\sum_{in} i_{i} - \sum_{out} i_{e} \nonumber \]

    In applying the conservation of charge equation, Equations \(\PageIndex{6}\) or \(\PageIndex{7}\), the approach is the same as we have used before. First you must clearly identify the system and think about the time period. Finite-time forms of the conservation of charge equation can be developed by integrating these rate forms of the conservation of charge equation with respect to time as done for mass. Be careful to clearly indicate the direction of all currents on your system diagram.


    This page titled 4.2: Conservation of Charge is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Donald E. Richards (Rose-Hulman Scholar) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.