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two points charges q and -q are separated by a distance 2l find the flux of electric field strength vector across the circle coinciding with the midpoint of line joining the two charges in the perpendicular plane

Asked by mahi905j | 22 Apr, 2022, 10:00: AM

Expert Answer

Figure shows the two charges +q and -q are separated by distance 2l .

Let a circle of radius R is passing through the mid point of line joining the charges at vertical plane.

Magnitude of electric field E₊ at a point P which is at a distance r from centre of circle is given as

begin mathsize 14px style E subscript plus space equals space fraction numerator q over denominator 4 pi epsilon subscript o end fraction cross times fraction numerator 1 over denominator left parenthesis space r squared plus l squared right parenthesis end fraction end style

Direction of above electric field is along the line joing the point P and charge +q as shown in figure

Similarly magnitude of electric field E_- at a point P which is at a distance r from centre of circle is given as

begin mathsize 14px style E subscript minus space equals space fraction numerator q over denominator 4 pi epsilon subscript o end fraction cross times fraction numerator 1 over denominator left parenthesis space r squared plus l squared right parenthesis end fraction end style

Direction of above electric field is along the line joing the point P and charge -q as shown in figure .

Resultant electric field E at a point P is given as

begin mathsize 14px style E subscript blank space equals space fraction numerator 2 q over denominator 4 pi epsilon subscript o end fraction cross times fraction numerator 1 over denominator left parenthesis space r squared plus l squared right parenthesis end fraction cos theta end style

begin mathsize 14px style E space equals space 2 space fraction numerator q over denominator 4 pi epsilon subscript o end fraction space cross times fraction numerator 1 over denominator left parenthesis space r squared plus l squared space right parenthesis end fraction cross times fraction numerator l over denominator square root of r squared plus l squared end root end fraction space equals space fraction numerator 2 q over denominator 4 pi epsilon subscript o end fraction cross times l over left parenthesis space space r squared plus l squared space right parenthesis to the power of 3 divided by 2 end exponent end style

Let us consider an infintesimal area dA = r dr dΦ at point P ( polar coordinate system )

Flux Φ of electric field throug the circle is given as

begin mathsize 14px style ϕ space equals space double integral E times d A space equals fraction numerator q l over denominator 4 pi epsilon subscript o end fraction space integral subscript 0 superscript 2 straight pi end superscript d ϕ integral subscript 0 superscript R fraction numerator 2 r d r over denominator open parentheses r squared plus l squared close parentheses to the power of 3 divided by 2 end exponent end fraction end style

begin mathsize 14px style ϕ space equals space fraction numerator q l over denominator 2 space epsilon subscript o end fraction space integral subscript 0 superscript R fraction numerator 2 space r d r over denominator open parentheses r squared plus l squared close parentheses to the power of 3 divided by 2 end exponent end fraction space equals space fraction numerator q l over denominator epsilon subscript o end fraction open parentheses 1 over l minus fraction numerator 1 over denominator square root of R squared plus l squared end root end fraction close parentheses end style

Answered by Thiyagarajan K | 22 Apr, 2022, 11:12: AM

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