Radio transmitter creats a time-dependentcurrent I(t)=I 0 sint in the radio ante

Question

   Radio transmitter creats a time-dependentcurrent I(t)=I0sint in the radio antenna, whereI0=20A and =6.3x106 persecond(corresponding to about the middle of the AM band). What isthe average power radiated by this antenna if it is 160mlong? Hint: EM radiation is produced by charge sloshing back andforth in the antenna. Argue that at any given instant I(t)=Qv/L,where Q is total charge involved in the sloshing, v is the driftspeed of the charges, and L is the antenna's length.    Radio transmitter creats a time-dependentcurrent I(t)=I0sint in the radio antenna, whereI0=20A and =6.3x106 persecond(corresponding to about the middle of the AM band). What isthe average power radiated by this antenna if it is 160mlong? Hint: EM radiation is produced by charge sloshing back andforth in the antenna. Argue that at any given instant I(t)=Qv/L,where Q is total charge involved in the sloshing, v is the driftspeed of the charges, and L is the antenna's length.

Explanation / Answer

this is a crude estimate (the actual calculation of power radiatedby antenna is quite involved). Power P radiated by an isolated charge Q with acceleration a is P = (Q^2a^2)/(6c^3) since I = Io sint = Qv/L, a = dv/dt = d/dt(LI/Q) = (L/Q) Io cost P = [(Q^2)/(6c^3)][(LIo/Q) cost ]^2 So time averaged power is

= [(Q^2)/(6c^3)](LIo/Q)^2 = [(LIo)^2/(12c^3)] = (160*6.3*10^6*20)^2*9*10^9/(3*27*10^24) = 45158 W

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