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Exercise:
Show that the potential difference between two pos in the field of a po charge Q is given by Delta V_AB k_C Q left fracr_B - fracr_A right where r_A and r_B are the distances of pos A and B from the po charge respectively.

Solution:
We ase that the pos A and B are on the same radial field line in the electric field of the po charge Q. The electric field as a function of the distance to the po charge is given by Er k_C fracQr^ 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 k_C q Q _r_A^r_B fracr^ textdr k_C q Q left -fracr right_r_A^r_B k_C q Q left fracr_A - fracr_B right The potential difference between pos A and B is Delta V_AB -fracW_Ato Bq -frack_C q Q left fracr_A - fracr_B rightq k_C Q left fracr_B - fracr_A right quad square The pos with the same distance to the po charge as B define an equipotential surface since they can all be reached from B through a circular arc with radius r_B. The work along a circular arc 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 po charge.
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Exercise:
Show that the potential difference between two pos in the field of a po charge Q is given by Delta V_AB k_C Q left fracr_B - fracr_A right where r_A and r_B are the distances of pos A and B from the po charge respectively.

Solution:
We ase that the pos A and B are on the same radial field line in the electric field of the po charge Q. The electric field as a function of the distance to the po charge is given by Er k_C fracQr^ 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 k_C q Q _r_A^r_B fracr^ textdr k_C q Q left -fracr right_r_A^r_B k_C q Q left fracr_A - fracr_B right The potential difference between pos A and B is Delta V_AB -fracW_Ato Bq -frack_C q Q left fracr_A - fracr_B rightq k_C Q left fracr_B - fracr_A right quad square The pos with the same distance to the po charge as B define an equipotential surface since they can all be reached from B through a circular arc with radius r_B. The work along a circular arc 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 po charge.
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Electrostatics
Tags
point charge, potential difference
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Difficulty
(2, default)
Points
0 (default)
Language
ENG (English)
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