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Bead Pull Measurements of Microwave Cavities  
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This frequency shift can be shown to be: %BEGINLATEX{label="freqshift" color="Black"}% \begin{displaymath*}  
Changed:  
< <  \left(\frac{\Delta f}{f}\right) = \frac{k}{4U}\int\int{\left(\mu H\left(x,y\right)^{2}  \epsilon E\left(x,y\right)^{2}\right)dx~dy}  
> >  \displaystyle\left(\frac{\Delta f}{f}\right) = \frac{k}{4U}\int\int{\left(\mu H\left(x,y\right)^{2}  \epsilon E\left(x,y\right)^{2}\right)dx~dy}  
\end{displaymath*} %ENDLATEX%  
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In the case of a circular bead of radius, r, and where the perturbation is small enough that k may be taken to be unity, %BEGINLATEX{label="freqshiftforbead" color="Black"}% \begin{displaymath*}  
Changed:  
< <  \left(\frac{\Delta f}{f}\right) = \left(\frac{\pi r^3}{U}\right)\left[\epsilon_0\left(\frac{\epsilon_r1}{\epsilon_r+2}\right)E^2 + \mu_0\left(\frac{\mu_r1}{\mu_r+2}\right)H^2\right]  
> >  \displaystyle\left(\frac{\Delta f}{f}\right) = \left(\frac{\pi r^3}{U}\right)\left[\epsilon_0\left(\frac{\epsilon_r1}{\epsilon_r+2}\right)E^2 + \mu_0\left(\frac{\mu_r1}{\mu_r+2}\right)H^2\right]  
\end{displaymath*} %ENDLATEX%  
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>  
Added:  
> >   StephenMolloy  17 Aug 2009  
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Bead Pull Measurements of Microwave Cavities  
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Where the fields are those before the perturbation, and the integral is over the volume of the probe. k is a geometrical factor to take account of the reorganisation of the fields around the perturbing object, and the electric and magnetic constants are those for the material of the probe.  
Changed:  
< <  If a dielectric material (μ = 0) is used, it can be seen that this technique can be used to investigate the electric field only. In addition, since the integral is over the volume of the probe, different shapes (e.g. a bead or a needle) may be used to measure the absolute value and direction of the field vectors.  
> >  In the case of a circular bead of radius, r, and where the perturbation is small enough that k may be taken to be unity,
Where E and H are the average values of the fields (since the perturbation is small enough that the fields may be viewed as constant over the small volume of the bead).
If a dielectric material (μ_r = 1) is used, it can be seen that the frequency shift is related to the electric field only, thus allowing the measurements to be totally independent of the magnetic field. In addition, since the integral is over the volume of the probe, different shapes (e.g. a bead or a needle) may be used to measure the absolute value and direction of the field vectors.  
Changed:  
< <  Thus, combinations of different materials (dielectric and magnetic) and shapes may be used to form a complete picture of the field structure within a cavity.  
> >  Thus, combinations of different materials (dielectric and metallic) and shapes may be used to form a complete picture of the field structure within a cavity.  
References  
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>
 
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Bead Pull Measurements of Microwave Cavities  
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Thus, combinations of different materials (dielectric and magnetic) and shapes may be used to form a complete picture of the field structure within a cavity.
References  
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< < 
 
> >  
Revision 207 Jul 2009  StephenMolloy
Revision 107 Jul 2009  StephenMolloy
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