Question

The total charge that has entered a circuit element is q(t) = 6 (1 - e-7t) when t ≥ 0 and q(t) = 0 when t < 0. The current in the element for t ≥ 0 can be represented as i left-parenthesis t right-parenthesis equals Upper B e Superscript negative a t A where B and a are real constants. Determine the values of B and a.

Answer 1

`A=42, B=-7`

Explanation:

The current function of time is defined as follows:

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

where
`q(t)` is the charge function.

For the given charge function of time
`q(t)=6\left( 1-e^(-7t)\right)` we have the following current function:

`I(t)=(d)/(dt) \left(6\left( 1-e^(-7t)\right)\right)=42e^(-7t)`

In the problem it is proposed that
`I(t)=Be^(-At)`.

Examining the expression of
`I(t)` we obtained by deriving
`q(t)` with the expression proposed by the problem and comparing term by term:

`I(t)=Be^(-At)=42e^(-7t)`

We conclude that
`A=-7` and
`B=42`.