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P(t) = V(t)I(t) derivation

  1. Jan 12, 2013 #1
    I kindly ask for assistance in derivation of the equation for instantaneous power in an electric circuit, P(t) = V(t) I(t). I want to derive it as rigorously as possible. Here's what I got:
    We start with [itex]P = {\bf F} \cdot {\bf v}[/itex], where [itex]{\bf v} = \frac{d\bf r}{dt}
    We know that the force exerted on a test charge q is given by [itex]{\bf F} = {\bf E} q[/itex], and for voltage we know [itex]dV = - {\bf E} \cdot dx[/itex].
    Inserting F in equation for power, we get [itex]P = {\bf E}q \cdot \frac{d\bf x}{dt} = {\bf E}\cdot dx \frac{q}{dt} = - dV \frac{q}{dt} .[/itex]
    How would I go from this, to the desired result, without taking a "quantum leap"?
    Is there a better way to actually derive this mathematically impeccably?
  2. jcsd
  3. Jan 12, 2013 #2
    You need first to decide whether you want the power in a circuit, as you described,

    or the flow of power through a single point in space, which you appear to be trying to calculate.

    The old fashioned definition of EMF made this quite clear for a circuit.

    The EMF of a circuit develops the power in that circuit, equal to the EMF times the current flowing in that circuit.
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