Question

The current through a wire depends on time as $i={{i}_{0}}+\alpha t$, where ${{i}_{0}}=10A$ and $\alpha =4A/s$. Find the charge crossed through a section of the wire in 10 seconds.

Answer 1

Hint: Write current as differential rate of flow of charge. Integrate the equation to obtain an expression for the total charge crossed through the section of wire in that time period. Substitute the given values to obtain the magnitude of charge crossed.

Complete answer:
Electric current is defined as rate of transference of charge. We can write electric current as
$I=\dfrac{dQ}{dt}$ where Q represents charge and t represents time.
This implies that,
$dQ=Idt$
The electric current through the wire is given as a function of time as $i={{i}_{0}}+\alpha t$. Substituting this value in above equation, we have
$dQ=({{i}_{0}}+\alpha t)dt$
To find the charge crossed through a section of the wire in 10 seconds, we integrate this equation from t = 0 to t = 10 sec. By doing so we get, charge crossed in given time interval
$Q=\int{dQ=\int\limits_{0}^{10}{Idt}}$
$Q=\int\limits_{0}^{10}{{{i}_{0}}dt+\alpha tdt}=\mathop{\left[ {{i}_{0}}t+\dfrac{\alpha {{t}^{2}}}{2} \right]}_{0}^{10}$
$Q=10{{i}_{0}}+\dfrac{\alpha {{(10)}^{2}}}{2}=10{{i}_{0}}+50\alpha $
It is given that ${{i}_{0}}=10A$ and $\alpha =4A/s$. Substituting these values, we have
$Q=10\times 10+50\times 4=300As$
Since, $1As=1C$ Therefore,
$Q=300C$
The charge crossed through a section of the wire in 10 seconds is 300 coulombs.

Additional Information:
The relation of current and voltage is given by ohm’s law which states that current through a conductor is linearly proportional to the voltage applied. Mathematically,
$V\propto I\Rightarrow V=IR$
The proportionality constant R is known as resistance of the conductor.
Electric current can be measured by an instrument known as ammeter.
Moving charges or electric current produces magnetic fields. It can be said that moving charges produce both electric and magnetic fields.

Note:
Electric current is defined as rate of transference of charge and can be written as $I=\dfrac{dQ}{dt}$
One ampere of current is the rate of flow of one coulomb of charge in one second.
Students must learn basic formulas of integration and differentiation as these are very useful in solving many problems.