Question

Current I through an inductor is increasing according to I=2t. variation of rate of increment of its energy (du/dt) with time t is correctly shown in graph :
(options are attached)​

Answer 1

Answer:

B is the correct answer.

Explanation:

In excepted three graphs, it is not followed

Answer 2

Given: Current through the inductor = I A

           Relation of current with time: I=2t

To Find: The graph showing the correct representation of $\frac{du}{dt}$ vs t.

Solution:

• For solving this question, we will use the formula for energy through an inductor:

                u = $\frac{1}{2}$ LI²                              (1)

where L is inductance in Henry

   and I is current through the inductor in Amperes

• In this case, I =2t

                  So, substituting the value of I in (1)

                     ⇒ u =$\frac{1}{2}$L(2t)²

                   ⇒ u = 2Lt²

• Now differentiating both sides by t

                      ⇒$\frac{du}{dt}$ = $\frac{d (2Lt^{2}) }{dt}$

                      ⇒$\frac{du}{dt}$ = 2L$\frac{d(t^{2}) }{dt}$              (2L is not differentiated as it is a constant)

                      ⇒$\frac{du}{dt}$ = 2L × 2t

                       ⇒$\frac{du}{dt}$ = 4Lt                        (2)

• If we compare equation (2) with y=mx + c,

We get:        $\frac{du}{dt}$ on y-axis

                   slope = 4L

                    t on x-axis

            and y-intercept is 0

So, the graph of $\frac{du}{dt}$ vs t will be a straight line passing through the origin.

This type of graph is seen in option C.

Therefore, option C. is the correct option.