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The current through a wire depends on time as $i = {i_0} + \alpha t$ where ${i_0} = 10A$ and $\alpha = 4A{s^{ - 1}}$. Find the charge crossed through a section of the wire in $10$ seconds.

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Last updated date: 20th Jun 2024
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Answer
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Hint: The charge is the property of particles that attract and repulse with each other when placed in an electrical field. The rate of flow of charges is called current. Applying the formula for charge integrates the equation using the given data to obtain the charge crossed through a section in the given time period.

Formula used:
$q = \int {idt} $
Here $q$ is the charge, $i$ is current, $t$ is the time.

Complete step by step answer:
The charge is the property of particles that attract and repulse with each other when placed in an electrical field. A charged particle exert force on each other. The square of the distance between two particles is inversely proportional to the force due to the charge. The SI unit of charge is the coulomb.

$1.6 \times {10^{19}}$ is the charge of a single electron. The electronic circuit uses an electric charge to do some useful work. The moving charges produce an electric current. The rate of flow of charges is called current. The SI unit of current is ampere. One ampere equals one coulomb per second. The current is measured in the circuit using an ammeter.
Charge crossed through a section of the wire
$q = \int {idt} $
Now integrate the above equation
$q = \int\limits_0^t {idt} $
Substitute the value of $i$in the above equation
$q = \int\limits_0^t {\left( {{i_0} + \alpha t} \right)dt} $
$q = {i_0}t + \alpha \dfrac{{{t^2}}}{2}$
Now substitute the values to obtain the charge
$q = 10 \times 10 + 4 \times \dfrac{{10 \times 10}}{2}$

$q = 300C$(Answer).

Note: Fundamental property that attracts or repulses each other when placed in the magnetic field is called charge. When the linear size of a charged body is smaller than the distance, then the size can be ignored and that charge body is called point charge. The process of giving charge to a body is called electrification. The charge experiences field is only in the electric field.