Energy in inductor and capacitor

[smslca]

Energy in inductor is given as (Li²])/2
but energy is power absorbed in t secs is E=integral(from 0 to t)Lidi here i is current.
since this integral stretches from 0 to t After doing the integral the current i must be turned into the time variable i.e E=(Lt²)/2;
Then why are we writing it as i² what is this i represent. Is this the current i and this Energy i the same or different.

Same problem for capacitance in terms of voltage

 

[Anamitra]

Power,P=Ei where E stands for the emf due to the inductive effect
Energy spent in time [0 to t]= integral[0 to t]Eidt

|E|= L di/dt

Energy expended= integral[0 to t]Li di/dt * dt
= integral[0 to i]Li di [please do note the change of variable here]
= 1/2 Li^2

 

[smslca]

[please do note the change of variable here]

since they may not be equal how could time change into current.
Let us suppose we want to find energy for 2 secs with current of x amps , and inductance 1 henry , then
IS E=0.5x joules using i as variable in boundary value correct or E=2 joules using t as variable in boundary value is a correct one.

 

[Anamitra]

Things would become clear if you know how current is changing with time i=i(t)

if you know the above function you can again find f(t) =di/dt
Then you find:

integral[0 to 2]L*i(t)* f(t)dt ------------- (1)

OR
You use the relation i=i(t) to calculate the current at time=2 seconds
that is you find i(2),noting i(0)=0

Then
Energy expended=
= integral[0 to i(2)]Li di
= 1/2 Li(2)^2 ---------------- (2)

(1) and (2) should give you the same result for any function i=i(t)provided they are differentiable and provided i(0)=0

 

[sardar]

The energy stored in an inductor and a capacitor is different and is calculated differently. In an inductor, the energy is stored in the form of a magnetic field, while in a capacitor, the energy is stored in the form of an electric field.

The equation E=(Li^2)/2 is the energy stored in an inductor, where L is the inductance and i is the current flowing through it. The integral equation E=integral(from 0 to t)Lidi represents the total energy absorbed by the inductor over a period of time t. The equation E=(Lt^2)/2 is derived from this integral equation by substituting the variable i with the time variable t.

Similarly, for a capacitor, the energy stored is given by E=(CV^2)/2, where C is the capacitance and V is the voltage across it. The integral equation for energy in a capacitor is E=integral(from 0 to t)Cvdv, where v is the voltage. This equation is also derived from the integral equation by substituting the voltage variable v with the time variable t.

In both cases, the variables i and v represent the current and voltage at a particular time, and the equations represent the total energy stored in the inductor or capacitor. So, the energy stored and the energy absorbed are the same, just calculated in different ways using different variables. It is important to note that the energy stored in an inductor or capacitor is not the same as the power absorbed, which is the rate at which energy is being transferred.

In summary, the equations E=(Li^2)/2 and E=(CV^2)/2 represent the energy stored in an inductor and a capacitor, respectively, and the variables i and v represent the current and voltage at a particular time. The integral equations represent the total energy absorbed over a period of time, which can be derived from the energy equations by substituting the appropriate variables.