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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 E=50x^i, where E is in NC1 and x is in metres.
Find
(i) Net flux through the cylinder.
(ii) Charge enclosed by the cylinder.
522015_4b83bb2bad2e43ac964070d413ed00b6.png

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Solution

Area of the cross-section of the cylinder A=25×104 m2
Electric field at the face nearer to the origin E1=20×0.5=10 N/C
Thus flux entering through this face ϕ1=E1A=10×25×104=2.5×102 Wb
Distance of second face from origin = 1+0.5=1.5 m
Electric field at the face away from the origin E2=20×(1.5)=30 N/C
Thus flux leaving out through this face ϕ2=E2A=30×25×104=7.5×102 Wb
(i) : Net flux through the cylinder ϕ=(7.52.5)×102=5×102 Wb
(ii) : Charge enclosed in the cylinder Q=ϕϵo
Q=5×102×8.85×1012=4.42×1013 C

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