Amplitude of field with phasor components

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To find the amplitude of a time-varying field represented by phasors, the instantaneous amplitude can be calculated using the real parts of the complex components: f = √(Re(X)² + Re(Y)² + Re(Z)²). The time-averaged amplitude is given by F_avg = √(X_rms² + Y_rms² + Z_rms²), where X_rms is derived from the complex conjugate of X. The total amplitude of the field is expressed as |F| = √(F* · F) = √(X*X + Y*Y + Z*Z), representing the magnitude in a six-dimensional space. This approach effectively utilizes phasor arithmetic similar to that in electronic circuits. Understanding these calculations is essential for analyzing time-varying fields.
misho
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Let's say I have a field (electric or magnetic or something) and it's time-varying so I choose to represent its components as phasors.

Say the field is:
\vec{F} = X\hat{x} + Y\hat{y} + Z\hat{z}

where, X, Y and Z are complex numbers.

Now, I want to find the amplitude of the field. If X, Y and Z were real (time constant field), I'd just go:

F = \sqrt{X^2 +Y^2 +Z^2}

but I have no idea what to do here. Also, I'm not sure if there's an easy way to do this or not. Any ideas?
 
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It works just like phasor arithmetic for electronic circuits. The instantaneous amplitude will be

f = \sqrt{\Re (X)^2 + \Re (Y)^2 + \Re (Z)^2}

The time-averaged amplitude is

F_{avg} = \sqrt{X_{rms}^2 + Y_{rms}^2 + Z_{rms}^2} = \sqrt{\frac{X^*X + Y^*Y + Z^*Z}{2}}

where X^* is the complex conjugate of X.

Lastly, the total amplitude of F is

|\vec F| = \sqrt{\vec F^* \cdot \vec F} = \sqrt{X^*X + Y^*Y + Z^*Z}

This is the magnitude of a vector in six-dimensional space, which rotates around on a 5-sphere such that its projections along the six axes are equal to the real and imaginary components of each of X, Y, and Z.

(Alternatively, you can think of F as living in three-dimensional complex space, such that its projection on each of the three complex axes gives the three complex numbers X, Y, and Z).
 
Thanks a lot! Answers my question perfectly.
 
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