Question

The coefficient of self induction of a coil is given by

$\left( a \right)L=\left( -\dfrac{dI}{dt} \right)$ 

$\left( b \right)L=-e\dfrac{dt}{dI}$ 

$\left( c \right)L=-\dfrac{dI}{edt}$ 

$\left( d \right)L=\dfrac{dI}{dt}{{e}^{2}}$ 

Answer 1

Hint: We will use the concept of Lenz's law. Since the formula of the law is similar to the definition of the self inductance of the coil, we will resemble both these definitions and find out that there is an opposition in the flow.

Formulas used:

$e.m.f.=-\left( \dfrac{ d\phi }{d t} \right)$ where e.m.f. is called the electromotive force. This e.m.f is equal to the difference in the flux with respect to the difference in time.

Complete step by step answer:

As the question talks about the self induction of a coil, we will first understand its induction property. By this property, we observe this in the conducting coil and when  the current changes in the conductor, then it  generates an induced  voltage in the form of electromotive  force. 

This polarity of induced emf is based on Lenz's law. By Lenz’s law, the direction of the induced emf is in such a way that it opposes the change in current which produces the induced emf.  

The formula of Lenz's law is $e.m.f.=-\left( \dfrac{ d\phi }{d t} \right)$, where e.m.f. is called the electromotive force. This e.m.f is equal to the difference in the flux with respect to the difference in time.  

Since, in this formula we can clearly see that there is opposition to the change in magnetic flux due to the presence of the negative sign, it will also result in the  opposition to the change of current by induced emf . By this formula of Lenz's law we find that the voltage generated opposes the change in the current. 

The inductance of a coil $L$ is related to the magnetic flux linked with the coil $\phi$ by the formula,

$\phi = LI$.....(1)

If we differentiate the above formula with respect to time, then

$ \dfrac{d\phi}{dt}=L\left( \dfrac{dI}{dt} \right) $ …….(2)

According to law of electromagnetic induction,

$e=-\dfrac{d\phi}{dt}$......(3)

Combining the equation (2) and (3),

$ -e=L\left( \dfrac{dI}{dt} \right) $

$ L=-\left( e\dfrac{dt}{dI} \right) $

Hence, the correct option is (B).

Note:

We will learn the following to solve the questions similar to this one.

(1) The negative sign in the formula of Lenz's law is important to note. As this will tell about the direction of the induced emf with the help of which we will find the direction of the current.

(2) The magnetic flux produced by a current carrying coil is directly proportional to the current passing through the coil.