An AC current is driven around a loop of wire. Suppose the amplitude and frequency of the current are both doubled. By what factor does the power radiated by the antenna increase?

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This is a magnetic dipole. A magnetic dipole in a vacuum radiates an average power by the following relation.

`ltPgt =(mu_0 m_0^2 omega^4)/(12 pi c^3)`

`mu_0` is the magnetic permeability of a vacuum, `m_0` is the average magnetic dipole moment, omega is the angular frequency, and `c` is the speed of light.

First we know that `m_0=Ia` , where `I` is the current and a is the area of the loop. We want to know `(< P_f >) / (< P_i >)` .

`(<P_f>)/(<P_i>)=(mu_0 I_f^2 a^2 omega_f^4)/(12 pi c^3)*(12 pi c^3)/(mu_0 I_i^2 a^2 omega_i^4)=(I_f omega_f^4)/(I_i omega_i^4)`

Plug in the changes between the final and initial configurations.

`(<P_f>)/(<P_i>)=(I_f^2 omega_f^4)/(I_i^2 omega_i^4)=((2I_i)^2 (2omega_i)^4)/(I_i^2 omega_i^4)=2^6`

The antenna radiates `2^6` times more power than initially.

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