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Exercise:
A cylindrical tank consists of conducting plates at the top and bottom and instulating side walls. It is partially filled with a liquid with dielectric constant kappa. The fill level can be determined by measuring the capacitance between the top and bottom plate. abcliste abc Derive a formal expression for the fill level as a fraction of the height of the tank as a function of the measured capacitance C and the capacitance of the empty tank C_. abc The empty tank has a capacitance CeO and is hO high. It is filled with benzene. The measured capacitance is CmO. Calculate the fill level. abcliste

Solution:
abcliste abc The partially filled tank can be erpreted as two capacitors in series. Their capacitances are C_ kappavarepsilon_fracAx h C_ varepsilon_fracA-x h where x is the fraction of the tank height filled with the liquid. It follows for the series capacitance sscCs leftfracC_+fracC_right^- leftfracx hkappavarepsilon_ A+frac-x hvarepsilon_ Aright^- C_leftfracxkappa+-xright^- C_leftfracx+kappa-xkapparight^- C_leftfracx-kappa+kappakapparight^- C_frackappax-kappa+kappa We have to solve this expression for x: sscCsleftx-kappa+kapparight C_kappa C_s x-kappa C_-C_skappa x resultfracsscCs-C_sscCsfrackappakappa- We can verify the solution for the extreme cases sscCsC_ empty tank and sscCskappa C_ full tank: sscxmin fracC_-C_C_frackappakappa- sscxmax frackappa C_-C_kappa C_frackappakappa- frackappa-kappafrackappakappa- Both results are as expected. abc The dielectric constant of benzene is kO. Using the solution from a we can express the fill level h_f as follows: h_f x h hfF fracCm-CeCmtimesfrackk-times h hf approx resulthfS abcliste
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Exercise:
A cylindrical tank consists of conducting plates at the top and bottom and instulating side walls. It is partially filled with a liquid with dielectric constant kappa. The fill level can be determined by measuring the capacitance between the top and bottom plate. abcliste abc Derive a formal expression for the fill level as a fraction of the height of the tank as a function of the measured capacitance C and the capacitance of the empty tank C_. abc The empty tank has a capacitance CeO and is hO high. It is filled with benzene. The measured capacitance is CmO. Calculate the fill level. abcliste

Solution:
abcliste abc The partially filled tank can be erpreted as two capacitors in series. Their capacitances are C_ kappavarepsilon_fracAx h C_ varepsilon_fracA-x h where x is the fraction of the tank height filled with the liquid. It follows for the series capacitance sscCs leftfracC_+fracC_right^- leftfracx hkappavarepsilon_ A+frac-x hvarepsilon_ Aright^- C_leftfracxkappa+-xright^- C_leftfracx+kappa-xkapparight^- C_leftfracx-kappa+kappakapparight^- C_frackappax-kappa+kappa We have to solve this expression for x: sscCsleftx-kappa+kapparight C_kappa C_s x-kappa C_-C_skappa x resultfracsscCs-C_sscCsfrackappakappa- We can verify the solution for the extreme cases sscCsC_ empty tank and sscCskappa C_ full tank: sscxmin fracC_-C_C_frackappakappa- sscxmax frackappa C_-C_kappa C_frackappakappa- frackappa-kappafrackappakappa- Both results are as expected. abc The dielectric constant of benzene is kO. Using the solution from a we can express the fill level h_f as follows: h_f x h hfF fracCm-CeCmtimesfrackk-times h hf approx resulthfS abcliste
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Electrostatics
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capacitance, capacitor, series circuit
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ENG (English)
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Calculative / Quantity
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