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A uniformly charged disc of radius R hav...

A uniformly charged disc of radius R having surface charge density `sigma` is placed in the xy plane with its center at the origin. Find the electric field intensity along the z-axis at a distance Z from origin :

A

`E=(2epsilon_(0))/(sigma)((1)/((Z^(2)+R^(2))^(1//2))+Z)`

B

`E=(sigma)/(2epsilon_(0))(1+(Z)/((Z^(2)+R^(2))^(1//2)))`

C

`E=(sigma)/(2epsilon_(0))(1-(Z)/((Z^(2)+R^(2))^(1//2)))`

D

`E=(sigma)/(2epsilon_(0))((1)/((Z^(2)+R^(2)))+(1)/(Z^(2)))`

Text Solution

Verified by Experts

The correct Answer is:
C
The disc can be considered to be a collection of large number of concentric rings. Consider an element of the shape of rings of radius r and of width dr. Electric field due to this ring at P is
`dE=(K.sigma2pir.dr.x)/((r^(2)+x^(2))^(3//2))`

Put, `r^(2)+x^(2)=y^(2)`
`2rdr=2ydy`
`therefore dE=(K.sigma2piy.dy.x)/(y^(3))=2Ksigmapi.x(ydy)/(y^(3))`
Electric field at P due to all rings is along the axis,
`therefore E=intdE`
`=2Ksigmapixunderset(x)overset(sqrt(R^(2)+x^(2)))int(1)/(y^(2))dy=2Ksigmapix.[-(1)/(y)]_(x)^(sqrt(R^(2)+x^(2)))`
`=2Ksigmapix[+(1)/(x)-(1)/(sqrt(R^(2)+x^(2)))]=2Ksigmapi[1-(x)/(sqrt(R^(2)-x^(2)))]`
`=(sigma)/(2epsilon_(0))[1-(x)/(sqrt(R^(2)+x^(2))]]` along the axis.
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