question_answer 30)

Obtain the formula for the electric field due to a long thin wire of uniform linear charge density without using Gauss's law.                 Hint. Use Coulomb's law directly and evaluate the necessary integral.  

Answer:

Consider a long thin wire of uniform linear charge density  placed along X-axis. Let P be a point lying on the y-axis                 From figure, and                 Change on an element AB is given by                                  Using Coulomb?s law, electric field at P due to this element is given by,                                                                                                                     ? (i)                 For a very long were, extends from to .                  Integrating,                                                                                                       ? (ii)                 where                 ? (iii)                 and                       ?. (iv)                 To solve (iii), put                                             .                 Also                 i.e.,                 Integrating with limitsand we get                                                                                                                                       ? (v)                 To solve equation (iv), Put                         or                                                                                                                                                ... (vi)                 Putting value of (v) and (vi) in equation (ii), we get .