Question

A current in a wire is given by the equation, . I=2t^(2)-3t+1, the charge through cross section of : wire in time interval t=3s to t=5s is

Answer 1

$\huge \underline {\underline{ \mathfrak{ \green{Ans}wer \colon}}}$

I = 2t² - 3t + 1

As we know that :

$\large{\boxed{\sf{I \: = \: \dfrac{dQ}{dt}}}} \\ \\ \implies {\sf{dQ \: = \: I dt}} \\ \\ \implies {\sf{dQ \: = \: (2t^2 \: - \: 3t \: + \: 1)dt}} \\ \\ \implies {\sf{\int dQ \: = \: \int (2t^2 \: - \: 3t \: + \: 1)dt}} \\ \\ \implies \displaystyle {\sf{Q = \int_{3}^{5} \: (2 t^{2} \: - \: 3t \: + \: 1)dt}} \\ \\ \implies {\sf{Q \: = \: \bigg[ \dfrac{2}{3} {t}^{3} \: - \: \dfrac{3}{2} {t}^{2} + t \bigg ]_{3}^{5}}} \\ \\ \implies {\sf{Q \: = \: \big[ \dfrac{2}{3} \: \times \: 98 \big] \: - \: \big [ \frac{3}{2} \: \times \: 16 \big] \: + \: 2}} \\ \\ \implies {\sf{Q \: = \: 65.33 \: - \: 24 \: + \: 2}} \\ \\ \implies {\sf{Q \: = \: 65.33 \: - \: 22}} \\ \\ \implies {\sf{Q \: = \: 43.33 \: C}}$

Answer 2

Answer:

Given:

I = 2t² - 3t + 1

This is the equation of current through.a conductor.

To find:

Charge flowing through the cross-section of the wire in the time interval of 3 - 5 seconds

Concept:

Current is defined as the rate of flow of charge.

Considering flow of charge in a very small time period , we can say that :

$\boxed{ \huge{ \red{ \bold{I = \dfrac{dq}{dt} }}}}$

Calculation:

From the equation , we can continue as :

$\boxed{ \green{ \huge{ \bold{ \displaystyle \int \: dq = \int \: I \: dt}}}}$

$\displaystyle = > q = \int(2 {t}^{2} - 3t + 1)dt$

Putting the limits , we get :

$\displaystyle = > q = \int_{3}^{5} \: (2 {t}^{2} - 3t + 1)dt$

$= > q = \bigg \{ \dfrac{2}{3} {t}^{3} - \dfrac{3}{2} {t}^{2} + t \bigg \}_{3}^{5}$

$= > q = (\frac{2}{3} \times 98) - ( \frac{3}{2} \times 16) + 2$

$= > q = 65.33 - 24 + 2$

$= > q = 43.33 \: coulomb$

So final answer :

$\boxed{ \blue{ \sf{ \huge{ \bold{q = 43.33 \: C}}}}}$