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Let us consider any surface *S*.
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If electrical charge passes through that surface
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we say that there is electrical current through that surface.
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Let us denote by *q*-sub-*AB*
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the time-dependent function which gives the charge that has crossed from *A* to *B*
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in each value of time *t*.
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And by "increment of *q*-sub-*AB*"
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to the charge that has crossed from *A* to *B*
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minus the one that has crossed from *B* to *A*
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in the same time interval of duration "increment of *t*".
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Thus, if for 3 seconds the charges that cross from *A* to *B*
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are 5 Coulombs and -7 Coulombs,
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and from *B* to *A* cross
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8 Coulombs and -4 Coulombs,
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the charge that crossed during that time from *A* to *B* is
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-6 Coulombs.
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And the "increment of *q*-sub-*BA*"
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is the charge that crosses at the same time from *B* to *A*
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minus the charge that crosses from *A* to *B*,
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6 Coulombs.
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The charge that crosses from *B* to *A* is always
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the opposite of the one that crosses from *A* to *B*.
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Therefore, saying that from *A* to *B*
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has crossed the charge "increment of *q*-sub-*AB*"
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means the same as saying that from *B* to *A*
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has crossed the charge "-increment of *q*-sub-*AB*".
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The quotient between the charge that crosses the *S* surface from *A* to *B*,
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and the time it takes to cross it,
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is called the *average current*
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*in that time interval from A to B*.
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Therefore, the average current *i*-sub-*AB* is:
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charge that, on average, crosses the surface *S*
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each time unit from *A* to *B*.
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Its unit in the *International System* turns out to be
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Coulomb per second, which is called Ampere,
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and its symbol is a capital A.
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As the elapsed time *t* is always positive,
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*i*-sub-*AB* always has the same sign as *q*-sub-*AB*.
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Also, since "increment of *q*-sub-*BA*"
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is the oposite of *q*-sub-*AB*
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it turns out that the current from *B* to *A*
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is opposite to the current from *A* to *B*.
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Thus, if in a range of 3 second
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the charge crossing from *A* to *B* is -6 Coulombs,
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the average current from *A* to *B*
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is -2 amperes,
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and from *B* to *A* it is 2 amperes.
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Therefore, saying that *i*-sub-*AB*
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is the current from *A* to *B*
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means exactly the same as saying that
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the current from *B* to *A* is "-*i*-sub-*AB*".
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As we reduce the time in which the average current is calculated,
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that average current approaches the current in an instant.
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This is why we call instantaneous current from *A* to *B*
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to the average current, when the time range approaches zero.
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That is to say, to the derivative of the charge *q*-sub-*AB* with respect to the time.
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As the charge *q*-sub-*BA* is the opposite of *q*-sub-*AB*,
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the instantaneous current from *B* to *A*
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is the opposite of the instantaneous current from *A* to *B*.
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For example, if the charge *q*-sub-*AB*
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is given in each time by the function
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Sine of (*Pi t*) minus 2,
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the instantaneous current in a given time *t* is
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*Pi* times the Cosine of (*Pi t*).
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Tha function gives the instantaneous current from *A* to *B*
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for any given value of *t*.
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Subtitules and English translation: Roberto C. Redondo Melchor