5 Complex Exponential Fourier Series Express the complex...

5 Complex Exponential Fourier Series Express the complex exponential Fourier series of v?(t) in Figure 1(a) using the complex exponential Fourier series of v?(t) in Figure 1(b). The complex exponential Fourier series of v?(t) in Figure 1(b) is given by $\infty$ $v_2(t) = \sum_{n=-\infty} D_n e^{jn\omega_0 t}$. Figure 1. (a) Periodic square pulse v?(t). Time in seconds. (b) Periodic square pulse v?(t). Time in milliseconds.

Transcript

00:01 Hello, in the question we have given that inertial frame s ' moves at a speed of 0 .81 c.

00:07 So this velocity of s ' is given that is 0 .81 c.

00:11 So with respect to the frame s, so we have this frame of reference.

00:16 So let us see what are the frame of reference.

00:18 So this is the frame s.

00:19 So this frame is s frame and over here we have this another frame of reference which we will call it as s'.

00:28 So this frame of reference is s'.

00:31 Now this frame of reference is moving with a velocity v is equal to 0 .81 c.

00:37 Now we will use the coordinate system.

00:39 So this is x, y and z.

00:41 So this is x, y and z coordinates.

00:44 So this is the origin.

00:45 Now what they are saying is they have given that at x is equal to x ' that is equal to 0, at t is equal to t ' that is equal to 0.

00:55 Now there are two events majored are recorded in the s frame.

01:01 So let us see what is event one.

01:04 So event one occurs at origin at t is equal to 0.

01:08 So it is x is equal to 0 at t is equal to 0.

01:12 Now if this is the condition, so now see what we have to do is we have to find according to the observer s', what is the time for event one and event two.

01:22 So this time is given by t ' is equal to gamma times t minus vx upon c square.

01:30 So now over here see now event two we haven't yet discussed what it is.

01:37 So we will come to that.

01:38 But event one is very simple.

01:40 So let us complete this first.

01:42 Now at x is equal to 0, t is 0.

01:44 So this term becomes 0.

01:46 So x is 0, t is 0.

01:48 So t ' will be 0.

01:49 And also they have given over here that t is equal to t ' is equal to 0.

01:57 From there we can directly write this.

01:59 Now the second event.

02:00 Let us come to the event two.

02:04 So event two occurs at on the x -axis at x is equal to 2 .3 kilometers.

02:12 It is 2 .3 kilometers.

02:20 So that is nothing but 2 .3 into 10 raised to 3 meters and time is equal to 5 .5 microseconds...

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