Potential Difference in Field of Charged Wire
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Exercise:
Show that the potential difference between two pos in the field of a charged wire with linear charge density lambda is given by Delta V_AB fraclambdapivarepsilon_ lnfracr_Ar_B where r_A and r_B are the distances of pos A and B from the wire respectively.
Solution:
We ase that the pos A and B are on the same radial field line in the electric field of the charged wire. The electric field as a function of the distance to the wire is given by Er fracpivarepsilon_ fraclambdar The work done by the electric field on a test charge q moving from A and B is W_Ato B _A^B vecF_E textdvecr _r_A^r_B q Er textdr fraclambdapivarepsilon_ q _r_A^r_B fracr textdr fraclambdapivarepsilon_ q left ln r right_r_A^r_B fraclambdapivarepsilon_ q left ln r_B - ln r_A right fraclambdapivarepsilon_ q ln fracr_Br_A The potential difference between pos A and B is Delta V_AB -fracW_Ato Bq -fracfraclambdapivarepsilon_ q ln fracr_Br_Aq -fraclambdapivarepsilon_ ln fracr_Br_A fraclambdapivarepsilon_ ln fracr_Ar_B quad square The pos with the same distance to the wire as B define an equipotential surface since they can all be reached from B with a combination of a circular arc with radius r_B or a displacement parallel to the wire. In each case the work is zero since the force is perpicular to the displacement. As a consequence the derivation is valid for any two pos A and B in the field of the wire.
Show that the potential difference between two pos in the field of a charged wire with linear charge density lambda is given by Delta V_AB fraclambdapivarepsilon_ lnfracr_Ar_B where r_A and r_B are the distances of pos A and B from the wire respectively.
Solution:
We ase that the pos A and B are on the same radial field line in the electric field of the charged wire. The electric field as a function of the distance to the wire is given by Er fracpivarepsilon_ fraclambdar The work done by the electric field on a test charge q moving from A and B is W_Ato B _A^B vecF_E textdvecr _r_A^r_B q Er textdr fraclambdapivarepsilon_ q _r_A^r_B fracr textdr fraclambdapivarepsilon_ q left ln r right_r_A^r_B fraclambdapivarepsilon_ q left ln r_B - ln r_A right fraclambdapivarepsilon_ q ln fracr_Br_A The potential difference between pos A and B is Delta V_AB -fracW_Ato Bq -fracfraclambdapivarepsilon_ q ln fracr_Br_Aq -fraclambdapivarepsilon_ ln fracr_Br_A fraclambdapivarepsilon_ ln fracr_Ar_B quad square The pos with the same distance to the wire as B define an equipotential surface since they can all be reached from B with a combination of a circular arc with radius r_B or a displacement parallel to the wire. In each case the work is zero since the force is perpicular to the displacement. As a consequence the derivation is valid for any two pos A and B in the field of the wire.