If electric current passing through a conductor varies with time as I = I₀ + βt, where I₀ = 20 A, β = 3 A/s, then find charge flow through conductor in first 10 sec.

The correct option is 350 C.

Step-by-step Derivation:

Electric current is defined as the rate of flow of electric charge. Mathematically, it is represented as:
$I=\frac{dq}{dt}$
where $dq$ is the small amount of charge flowing in an infinitesimal time interval $dt$.

To find the total charge $q$ flowing through the conductor in a given time interval, we rearrange the equation and integrate both sides:
$dq=Idt$
$q={\int }_{t_1}^{t_2}Idt$

We are given:
- The current varying with time: $I=I_0+\beta t$
- Initial current parameter, $I_0=20\text{A}$
- Constant coefficient, $\beta =3\text{A/s}$
- Time interval: from $t=0\text{s}$ to $t=10\text{s}$

Substituting the expression for current $I$ into the integral:
$q={\int }_0^{10}(I_0+\beta t)dt$

Integrating the terms with respect to $t$:
$q={[I_{0}t+\frac{1}{2}\beta t^2]}_0^{10}$

Now, evaluate the limits:
$q=(I_0·10+\frac{1}{2}\beta ·{10}^2)-0$
$q=10I_0+50\beta$

Substitute the given numerical values of $I_0$ and $\beta$:
$q=10\left(20\right)+50\left(3\right)$
$q=200+150$
$q=350\text{C}$

Therefore, the charge flow through the conductor in the first 10 seconds is 350 C.