Question

A sinusoidal voltage $V(t)=100\sin (500t)$ is applied across a pure inductance of $L=0.02H$. The current through the coil is:
$\begin{align}
  & A)10\cos (500t) \\
 & B)-10\cos (500t) \\
 & C)10\sin (500t) \\
 & D)-10\sin (500t) \\
 & \\
\end{align}$

Answer 1

Hint: For a pure inductor like a coil of wire, voltage across the inductor is proportional to the rate of change of current through the inductor with respect to time. The constant of proportionality in this case is termed as inductance, which is the property of a coil to resist any change in current flowing through the conductor. Current flowing through a pure inductor can easily be determined by taking the advantage of integration, as can be seen in the following solution.
Formula used:
$V(t)\propto \dfrac{dI}{dt}=L\dfrac{dI}{dt}$

Complete answer:
We know that voltage across a pure inductor like a coil of wire is proportional to the rate of change of current through the coil. If $V(t)$ represents the sinusoidal voltage applied across an inductor, then,
$V(t)\propto \dfrac{dI}{dt}=L\dfrac{dI}{dt}$
where
$V(t)$ is the sinusoidal voltage applied across pure inductor like a coil of wire
$\dfrac{dI}{dt}$ is the rate of change of current, flowing through the coil with respect to time
$L$ is the constant of proportionality termed as inductance of the coil
Let this be equation 1.
Coming to our question, we are provided with a sinusoidal voltage $V(t)=100\sin (500t)$, being applied across a pure inductance of $L=0.02H$. We are required to determine the current through the coil.
Substituting the given values in equation 1, we have
$V(t)=L\dfrac{dI}{dt}\Rightarrow 100\sin (500t)=0.02\times \dfrac{dI}{dt}$
where
$V(t)=100\sin (500t)$ is the voltage applied across the coil, as provided in the question
$L=0.02H$ is the inductance of the coil, as provided
Rearranging the above expression, we have
\[100\sin (500t)=0.02\times \dfrac{dI}{dt}\Rightarrow dI=\dfrac{100\sin (500t)\times dt}{0.02}\]
Let this be equation 2.
Now, to determine the total current flowing through the coil, let us integrate equation 2 as follows:
\[\int{dI}=\int{\dfrac{100\sin (500t)\times dt}{0.02}}\Rightarrow I=\dfrac{100}{0.02}\int{\sin (500t)dt}\Rightarrow I=\dfrac{-100\cos (500t)}{0.02\times 500}=-10\cos (500t)\]
Let this be equation 3.
Therefore, from equation 3, we can conclude that current flowing through the given coil is equal to $-10\cos (500t)$.

Hence, the correct answer is option $B$.

Note:
Inductance of a coil is the tendency of the coil to resist any changes in the current, flowing through the coil. Its SI unit is $henry(H)$. One $henry(H)$ is defined as the inductance which causes a voltage of $1V$ across an inductor, when current passes through the inductor at the rate of $1A$ per second $(s)$. When current flows through a pure inductor, a magnetic field is created around the inductor. The current flowing through the inductor adjusts with this magnetic field. The generated magnetic field also causes an electromotive $(emf)$ voltage across the inductor, due to electromagnetic radiation.