Energy of a wave in a transmission line

[walking_edges]

Homework Statement

An ideal transmission line has a characteristic impedance of Z=50 Ohms and
v=200,000km/s propagation velocity. A sinusoidal signal with frequency f=1GHz and
A=10mV amplitude is traveling down the line.Its total duration is 10s.What total
energy is it carrying?

Homework Equations

P = IV
w = 2[pi]f
k=w/v
V(x,t)=Re{A exp[i(wt-kx)]}
I(x,t)=Re{A/Z exp[i(wt-kx)]}

The Attempt at a Solution

I drew a picture of a transmission line that terminated into to a resistor with the same characteristic impedance as the line. The sinusoidal wave was coming in from the left only. The reflection amplitude of this wave is 0, so I think I should be able to say that all the energy in the wave was "burned" off by the resistor.

A resistor burns energy at the rate of V^2/R J/s, which I said was equal to (A)^2/(Z), the amplitude of the wave squared divided by the characteristic impedance. I then multiplied the power by 10 seconds to get the total energy dissipated by the resistor, which I calculated to be 2*10^-5 J.

The small answer and the fact that I didn't use all the information given pretty much yells to me that I did something wrong. I imagine that time and position dependence of the voltage and current might have something to do with it, but I'm not exactly sure.

 

[anathema]

Thank you for your post. Your approach to the problem is correct, but I believe you may have missed a key concept in calculating the total energy carried by the sinusoidal wave.

While it is true that the resistor dissipates all the energy in the wave, the wave itself continues to travel down the transmission line. This means that the total energy carried by the wave is not just the energy dissipated by the resistor, but also the energy that is still present in the wave as it continues to propagate.

To calculate the total energy, you will need to integrate the power over the entire duration of the wave's propagation. This can be done using the equations you have provided, as well as the fact that the total duration of the wave is 10 seconds.

I hope this helps clarify the solution for you. If you have any further questions, please let me know.