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Question
A hollow cylindrical box of length 1 m and area of cross-section 25 cm2 is placed in a three dimensional coordinate system as shown in the figure. The electric field in the region is given by `vecE = 50xhati` where E is NC−1 and x is in metres. Find
(i) Net flux through the cylinder.
(ii) Charge enclosed by the cylinder.
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Solution
Given,`vecE = 50xhati and Δs = 25 cm^2 = 25 xx 10^-4 m^2`
As the electric field is only along the x-axis, so, flux will pass only through the cross-section of cylinder.
magnitude of electric field at cross - section A,`E_A = 50 xx 1 =50N C^-1`
magnitude of electric field at cross - section `B,E^B = 50 xx 2 = 100 N C^-1`
The corresponding electric fluxes are :
`ø_A = vecE.Δvecs = 50 xx 25 xx 10^-4 xx cos 180° = -0.125 N m^2 C^-1`
`ø_A = vecE.Δvecs = 100 xx 25 xx 10^-4 xx cos 0° = 0.25N m^2 C^-1`
So, the net flux through the cylinder,`ø =ø_A +ø_B = -0.125 + 0.25 = 0 . 125 N m^2C^-1 `
(ii) Using Gauss’s law: `ointvecE.dvecs =q/in_0⇒ 0.125 = q/(8.85 xx 10^-12)⇒ q=8.85 xx 0.125 xx 10^-12 =1.1 xx 10 ^-12 C `
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