3) a) Consider a uniformly charged thin rod (line charge) of length L with a pos

Question

3) a) Consider a uniformly charged thin rod (line charge) of length L with a positive linear charge density . The rod is aligned along z and the centre of the rod is the origin of the coordinate system. Find the electric field at a point P at a distance y from the rod along its perpendicular bisector (the positive y axis) as shown in Figure 26-6. You must show all steps in the calculation and complete the integration without using integral tables. Hint: Use the substitution z =y sinh t and 1+sinh't coshr 2 b) Assume that the line charge extends from -oo to +oo along the z-axis. Calculate the electric field at a point P on the basis of your result in part a)

Explanation / Answer

3) a. given uniformly charged thin rod of length L, charge density lambda
   now, consider a charge element of length dz on the rod at distance z from the center of the rod
   electric field due to this charge at distance y on the perpendicular bisector of the rod is given by
   dE = k*dq/(z^2 + y^2)
   but dq = lambda*dz
   dE = k*lambda*dz*y/(z^2 + y^2)^3/2
   the sin component of the electric field will be cancelled as the point is symmetrical about the perpendicular bisector of the rod
   integrating it from z = -L/2 to z = +L/2
   E = k*lambda((L/2)/(L^2/4 + y^2)^1/2 + (L/2)/(L^2/4 + y^2)^1/2)/y
   E = k*lambda/y*(1/4 + y^2/L^2)^1/2

   b. when L -> infinity

y/L = 0
   E = 2k*lambda/y

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