Find radiated energie using Parseval Theorem

[tsumi]

Homework Statement

A radiated wave has electrical field:

E = e^-t/τ sen(ω0t) for t>0 and 0 for t<0

Using Parseval Theorem calculate the radiated energie in the intervale of frequencies: (ω,ω+dω)

Homework Equations

Parseval Theorem: ∫ |E(t)|^2 dt = (1/2π) ∫ |E(ω)|^2 dω

The Attempt at a Solution

I am not sure about what is demanded. Am I simply supposed to find E(ω)? To do so I make E(t).E*(t) to get |E(t)|^2, integrate, and then how do I get to E(ω)?

Thank you for any attention

 

[mighty2000]

.Hello,

To use Parseval's theorem in this case, you need to first find the Fourier transform of the given electric field function, E(t). The Fourier transform is defined as:

E(ω) = ∫ E(t) e^(-iωt) dt

Once you have the Fourier transform, you can use Parseval's theorem to calculate the total radiated energy in the given interval of frequencies. The equation you provided is correct, but it is missing a factor of 2π in the denominator.

So, the correct equation using Parseval's theorem would be:

∫ |E(t)|^2 dt = (1/2π) ∫ |E(ω)|^2 dω

where E(ω) is the Fourier transform of E(t).

To find E(ω), you can use the definition of the Fourier transform and the given electric field function. You can also use the fact that the Fourier transform of a product of two functions is equal to the convolution of their individual Fourier transforms.

I hope this helps. Let me know if you need further clarification.