Energy absorbed by the inductor in 4 seconds

[jaus tail]

Homework Statement

Homework Equations

V = L di/dt
Energy = V*I*t

The Attempt at a Solution

First we find out V across L
This V will be there only for 1st 2 seconds as after that there is no change in current.
So V(across L) is L * di/dt = 2 * 6/2 = 6V equation 1
Now Current is a function of time so it's i = 3t equation 2
So to get Energy we use integration.
Energy = integration of V*I*dt where limits are 0 to 2
So now using equation 1 and 2 we get
Energy = integration of 6 * 3t dt from limits 0 to 2
so this is 18 t²/2 t varies from 0 to 2
So this is 9 (4) = 36 which is not in the options.
Where am I wrong.
I even tried other formula of Energy = 1/2 * L * I² But this gives 24
I'm getting different answer for different formula but neither answer matches options. Where am I wrong?

 

[gneill]

I don't see where you've taken into account the resistance of the inductor. Sure, the inductor won't be storing the energy lost to heat via the resistance, but it's energy absorbed (and then dissipated) by the inductor nonetheless.

I note that you didn't include the general formula for energy stored in an inductor in your relevant equations. It might prove handy in this instance

 

[Windadct]

The question is a little ambiguous - but I do think they are looking for total energy, Inductance (stored) and Resistance ( dissipated)

 

[jaus tail]

Book says answer is C
But I don't know how they got that. Power in inductor is 1/2 * L * i²
Why do I have to include resistance? The current is given. So current through inductor is known.

W = 1/2 L*i*i
Energy is integrating W*dt as t varies from 0 to time asked in question.

 

[gneill]

Why do I have to include resistance?

The inductor, as a real component, has both inductance and resistance due to the resistivity of the coil of wire that it is made from. That wire resistance will dissipate energy in addition to the energy that the inductance will store in its magnetic field.

 

[jaus tail]

Oh okay. So I'll have to integrate i²R with respect to dt as t varies from 0 to 4 as well. Got it. Thanks.
That gives Energy across L as 36
Energy across R for transient time is 24
Energy across R in steady state is 72
Adding all I get answer as a)
Thanks.

 

[Windadct]

Also -- in this question the current is "set", formally defined -- normally (in a real case) both the inductor and the resistance would be impacting the total current and you would be seeing the impact of both elements more clearly..