An infaite lare stralg ht Line, chorgrdwith Lineor desity chofge , (oincid! wit
ID: 1884377 • Letter: A
Question
An infaite lare stralg ht Line, chorgrdwith Lineor desity chofge , (oincid! wit h the X axis ot the systrms Find the Fields nd tmS u ih vloty V in t hex" posiivc oxis the xpresiom For with the orresponding ot an x" posihc oxis direcdion Compere wit h conductor wire and Show +hot they mochted Notice tcro in the movement sysem ercn ctween an co adu whot s the physire! diFereare betucn on coadoc to si Ca :(e and a line charged moving in the direction °+ his eng At , thot could exploin the oppcarana nc chorgeExplanation / Answer
given infinite long straight wire
linear charge density lambda, coincides with x axis of the system
consider an axes system with x axis parallel to this systems x axis but relative speed v in this direction
then
consider a small charge dq on the wire for length dx
magneitc field of a moving charge is given by
dB = k*d*vdq/r^3 [ where d is perpendicular distance of the point of observation in moving axis to the infinitely long charged wire]
now
dq = lambda*dx
r = sqrt(x^2 + d^2)
dB = k*d*v*lambda*dx/(x^2 + d^2)^3/2
integrating from -L/2 to +L/2
B = 2k*lambda*v/d
electric field can be directly found from gauss law
E= 2k'*lambda/d
for a conducting wire carrying current I in the rest frame of refernce, and f the point of observation is also in the same frame
then
B = 2kI/d
but
from the magnetic field in the moving gase
lambda*v = (dq/dx)*dx/dt = dq/dt = I
hence botht he formulas are identical
E' is different for line charge and current in conductor case because in current passing throuigh conductor the conductor on the whole stays neutral where as in this case the conductor is not neutral and hence there is E' appearing