412 MAGNETIC ACTION ON LIGHT. [829.
magnetic force, what we want to determine is the value of ---, when n
d y
is constant, in terms of -~ , when y is constant. Differentiating (1 8) aw/
^ 0. (19)
We thus find = - -L - (20)
ay pn
��829.] If A is the wave-length in air, and i the corresponding index of refraction in the medium,
q\ = 2-ni, n\ = 2nv. (21)
The change in the value of q, due to magnetic action, is in every case an exceedingly small fraction of its own value, so that we may
write g
(22)
��where q Q is the value of q when the magnetic force is zero. The angle, 0, through which the plane of polarization is turned in passing through a thickness c of the medium, is half the sum of the positive and negative values of qc, the sign of the result being changed, because the sign of q is negative in equations (14). We thus obtain
- =- Cr | (23)
\isC i 2 ,. di^ 1
--^T^O^sx) - (24)
1 27r(?y -
vp\
The second term of the denominator of this fraction is approx imately equal to the angle of rotation of the plane of polarization during its passage through a thickness of the medium equal to half a wave-length. It is therefore in all actual cases a quantity which we may neglect in comparison with unity.
Writing ~ = m, (25)
we may call m the coefficient of magnetic rotation for the medium, a quantity whose value must be determined by observation. It is found to be positive for most diamagnetic, and negative for some paramagnetic media. We have therefore as the final result of our theory ^2 .7;
i-K, (26)
��where 6 is the angular rotation of the plane of polarization, m a
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