Dipole in Field of Charged Wire
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Exercise:
A dipole is placed in the field of a uniformly charged wire linear charge density lambda with its dipole moment vec p q vec d ed in the direction of the field lines. center includegraphicswidthtextwidth#image_path:dipoland-wir# center abcliste abc Show that for a distance r from the po charge Q with r gg d the force acting on the dipole can be written as F - fracpivarepsilon_ fraclambda pr^ abc Calculate the force on an HF hydrogen fluoride molecule at a distance rO from a wire with a linear charge density laO. The dipole moment of an HFL molecule is pO. Compare the result to the force acting on an electron at the same distance. abcliste
Solution:
abcliste abc The forces on the positive and negative charge of a dipole at distance r from a po charge Q are F_+ q E_Qr+d/ q fracpivarepsilon_ fraclambdar+d/ F_- -q E_Qr-d/ -q fracpivarepsilon_ fraclambdar-d/ where we ase a simple dipole with its negative charge facing the positively charged wire see figure. The net force is thus F q fracpivarepsilon_ lambda leftfracr+d/ - fracr-d/ right fracq lambdapivarepsilon_ fracr-d/ - r+d/r+d/ r-d/ fracq lambdapivarepsilon_ frac-dr^-d^/ &approx fracq lambdapivarepsilon_ frac-dr^ - fracpivarepsilon_ fracq d lambdar^ - fracpivarepsilon_ fraclambda pr^ In this situation the force pos towards the wire negative sign. abc The force acting on the molecule is F FF fracpitimesnceps timesfraclatimespr^ resultFS The force on an electron at the same distance from the wire would be F' FeF fracpitimesnceps times fraclatimesncer resultFeS The force on the dipole is much weaker than that on the electron. abcliste
A dipole is placed in the field of a uniformly charged wire linear charge density lambda with its dipole moment vec p q vec d ed in the direction of the field lines. center includegraphicswidthtextwidth#image_path:dipoland-wir# center abcliste abc Show that for a distance r from the po charge Q with r gg d the force acting on the dipole can be written as F - fracpivarepsilon_ fraclambda pr^ abc Calculate the force on an HF hydrogen fluoride molecule at a distance rO from a wire with a linear charge density laO. The dipole moment of an HFL molecule is pO. Compare the result to the force acting on an electron at the same distance. abcliste
Solution:
abcliste abc The forces on the positive and negative charge of a dipole at distance r from a po charge Q are F_+ q E_Qr+d/ q fracpivarepsilon_ fraclambdar+d/ F_- -q E_Qr-d/ -q fracpivarepsilon_ fraclambdar-d/ where we ase a simple dipole with its negative charge facing the positively charged wire see figure. The net force is thus F q fracpivarepsilon_ lambda leftfracr+d/ - fracr-d/ right fracq lambdapivarepsilon_ fracr-d/ - r+d/r+d/ r-d/ fracq lambdapivarepsilon_ frac-dr^-d^/ &approx fracq lambdapivarepsilon_ frac-dr^ - fracpivarepsilon_ fracq d lambdar^ - fracpivarepsilon_ fraclambda pr^ In this situation the force pos towards the wire negative sign. abc The force acting on the molecule is F FF fracpitimesnceps timesfraclatimespr^ resultFS The force on an electron at the same distance from the wire would be F' FeF fracpitimesnceps times fraclatimesncer resultFeS The force on the dipole is much weaker than that on the electron. abcliste