# The current in a wire varies with time according to the relation I=55A?(0.65A

The current in a wire varies with time according to the relation $I=55A?\left(0.65\frac{A}{{s}^{2}}\right){t}^{2}$.
How many coulombs of charge pass a cross section of the wire in the time interval between $t=0$ and $t=8.5s$ ?
What constant current would transport the same charge in the same time interval?

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Step 1
The charge passing through a cross section in the time interval is given by
$q={\int }_{{t}_{1}}^{{t}_{2}}Itd$
Substitute $\left(55A-\left(0.65\frac{A}{{s}^{2}}\right){t}^{2}\right)$ for I, 0s for ${t}_{1}$, and 8.5s for ${t}_{2}$ to find the charge.
$q={\int }_{0s}^{8.5s}\left(55A-\left(0.65\frac{A}{{s}^{2}}\right){t}^{2}\right)dt$
$={\left[55tA\right]}_{0s}^{8.5s}-{\left[\frac{0.65{t}^{3}}{2}\right]}_{0s}^{8.5s}$
$=467.5C-133.1C$
$=334.1C$
The amount of charge that pass a cross section in the given time interval is 334.1C.
Step 2
The constant current is given by the equation
$I=\frac{Q}{\mathrm{△}t}$
$=\frac{Q}{{t}_{2}-{t}_{1}}$
Substitute 334.1 C for Q and 8.5s for ${t}_{2}$ and 0s for ${t}_{1}$ to find I.
$I=\frac{334.1C}{8.5s-0s}$
$=39.3A$.
The constant current that would transport the charge is 39.3A.
The amount of charge that pass a cross section in the given time interval is 334.1C.
The constant current that would transport the charge is 39.3A.