a) Use Gauss\'s law to determine Din the region indicated for the following char

Question

a) Use Gauss's law to determine Din the region indicated for the following charge distributions. b) Qualitatively explain the behavior of the field as a function of distance from the charge distribution Outside an infinitely-long line charge with uniform line charge density .(Compare the result to the corresponding result of Coulomb's law). Outside a spherical surface of radius a and uniform surface charge density Outside an infinitely-long cylindrical surface of radius a and uniform surface charge density ps . Outside an infinite sheet of uniform surface charge density that lies in the z 0 plane

Explanation / Answer

a. to find electric field intensity

for infinitely long line charge with uniform line charge density rhol

consider a cylinderical surface around the line charge at dsistance r with the axis of the chylinder be the line charge

then charge contained inside the volume, q = rhol*l ( where l is length of the cylinder)

from gauss law

2*pi*r*l*E = rhol*l/epsilon

E = rho/2*pi*epsilon*r

hence the electric field falls radially with distance and is cylindrically symmetric

b. for a uniformly charged spherical surface of radius a and uniform charge density rhos

tyhe electric field will be spherically symmetriuc hence consider a spherical gaussean surface concentric with the sphere

then

from gauss law

4*pi*r^2*E = rhos*4*pi*a^2/epsilon

then

E = rhos*a^2/e^2*epsilon

hence

electidc field falls off with square of distance from the spherer center and is sphertically symmetric

c. outside cylindrical surface of charge density rhos, radius a

we can have the same case as part 1

2*pi*r*l*E = rhos*2*pi*a*l/epsion

E = rhos*a/r*epsilon

hence the electirc field falls of with the radial distance form the axis of the cylinder

d. outisde infinite charged sheet, charge density rhos

consider a cylinder of length 2l, with circular faces parallel to the plane of the sheet and sheet in the mid seciton of the cylinder

hence

from gauss law

2*pi*r^2*E = 2*pi*r^2*rhos/epsilon

E = rhos/epsilon

hence the lectiec vield remains constant with distance form the sheet

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