Question

The current through a wire depends on time as $I=(2+3t)\text{A}$. Calculate the charge crossed through a cross-section of the wire in $10\text{s}$.

• A. $150\text{C}$
• B. $17\text{C}$
• C. $20\text{C}$
• D. $170\text{C}$

Answer 1

The correct option is D $170\text{C}$

Given:
$I=(2+3t)\text{A}$

As we know that

$I=\frac{dq}{dt}\Rightarrow dq=Idt$

$\Rightarrow dq=(2+3t)dt$

Integrating both side,

$\Rightarrow \underset{0}{\overset{q}{\int }}dq=\underset{0}{\overset{10}{\int }}(2+3t)dt$

$\Rightarrow q={(2t+\frac{3t^2}{2})}_0^{10}=2(10-0)+\frac{3}{2}\times (100-0)$

$\Rightarrow q=20+150=170\text{C}$

So, option $\left(a\right)$ is right.

Why this Question ? The current at time $t$ is $$I=\underset{\Delta t\to 0}{lim}\frac{\Delta Q}{\Delta t}=\frac{dQ}{dt}$$ Thus, electric current through an area is the rate of transfer of charge from one side of the area to the another.