21.7 Applications of calculus by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.

**vCalc Companion Formulas** |

**vCalc Formulary** |
**21.7 Application of calculus** |

`I=(dq)/(dt)` |
Electric current (not constant) |

## 21.7 Applications of calculus (optional calculus-based section)

As discussed in example 1 on page 568, the definition of current as the rate of change of charge with respect to time must be reexpressed as a derivative in the case where the rate of change is not constant,

`I=(dq)/(dt)`.

##### Example 9: Finding current given charge

? A charged balloon falls to the ground, and its charge begins leaking off to the Earth. Suppose that the charge on the balloon is given by `q=ae^(?bt)`. Find the current as a function of time, and interpret the answer.

? Taking the derivative, we have

`I=(dq)/(dt)`

`=?abe^(?bt)`

An exponential function approaches zero as the exponent gets more and more negative. This means that both the charge and the current are decreasing in magnitude with time. It makes sense that the charge approaches zero, since the balloon is losing its charge. It also makes sense that the current is decreasing in magnitude, since charge cannot flow at the same rate forever without overshooting zero.

21.7 Applications of calculus by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.