Amplitude of field with phasor components

[misho]

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?

 

[Ben Niehoff]

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).

 

[misho]

Thanks a lot! Answers my question perfectly.