When a current in a certain inductor coil is $ 5.0A $ and is increasing at the rate of $ 10.0\dfrac{A}{s} $ the potential difference across the coil is $ 140V $ . When the current is $ 5.0A $ and decreasing at a rate of $ 10.0\dfrac{A}{s} $ , the potential difference is $ 60V $ . Find the resistance of the coil.

Hint: When an electric current flows through an inductor, it stores energy in the form of a magnetic field. It is also known as a coil, chokes, or reactor. Inductors are generally made up of an insulated wire coiled into a coil. The inductance of an inductor is defined as the ratio of the voltage to the rate of change of current.
We have a relationship that is, the relation between voltage, current, and inductance as,
 $ V = L\dfrac{{di}}{{dt}} + iR $ ,
where $ V = $ voltage; $ L = $ inductance; $ R = $ resistance; $ t = $ time; $ i = $ current.

Complete Step By Step Answer:
Here we have asked to find the resistance of the coil when the current rate is increasing and decreasing. From Ohm's law, we know the relation between voltage and current as,
 $ V = i \times R $ …………(1)
In this problem, not only the resistance of the coil but also the inductance resistance need to be considered while calculating voltage.
 $ V = L\dfrac{{di}}{{dt}} $ ……………(2).
 Combining equations (1) and(2) we get
 $ V = L\dfrac{{di}}{{dt}} + iR $ …………….(3)
Given that,
For the first case of increasing the current rate,
 $ \dfrac{{di}}{{dt}} = + 10.0\dfrac{A}{s} $ ;
 $ i = 5A $
 $ V = 140V $ .
 Substituting these on (3) we get,
 $ 140 = \left( {L \times 10} \right) + 5R $ ………..(4)
For the second case of decreasing the current rate,
 $ \dfrac{{di}}{{dt}} = - 10A $ ;
 $ i = 5A $ ;
 $ V = 60V $ .
Substituting in (3) we get,
 $ 60 = - 10L + 5R $ ………………(5)
Solving equations (4) and(5) by subtracting both we get,
 $ 140 - 60 = 20L $
 $ \Rightarrow 80 = 20L $
 $ \Rightarrow L = 4H $
where $ H = $ Henry SI unit of Inductance $ \left( L \right) $
 By adding (4) and (5) we get,
 $ 200 = 10R $
 $ \Rightarrow R = 20\Omega $
So the resistance of the coil becomes, $ R = 20\Omega $ .

Note:
Faraday's law of induction is an electromagnetism fundamental law that predicts how a magnetic field would interact with an electric circuit to create an electromotive force (EMF)—a process known as electromagnetic induction. Transformers, inductors, generators, and solenoids, etc work on this principle. It asserts that EMF exists on the conductive loop when the magnetic flux across the surface encompassed by the loop fluctuates over time.