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Question: A cylinder is charged by 10µC. Length of cylinder is 1 km and radius Surface density of charge of cy...

A cylinder is charged by 10µC. Length of cylinder is 1 km and radius Surface density of charge of cylinder is

A

1.59 x 10⁻⁴ C/m²

B

1.59 x 10⁻² C/m²

C

1.59 x 10⁻³ C/m²

Answer

1.59 x 10⁻⁴ C/m²

Explanation

Solution

For a uniformly charged cylindrical surface the surface charge density is given by

σ=QArea\sigma=\frac{Q}{\text{Area}}

Since the charge QQ is distributed uniformly over the curved (lateral) surface of the cylinder, the area is

A=2πrL,A=2\pi rL,

where rr is the radius and LL is the length of the cylinder.

Thus,

σ=Q2πrL.\sigma=\frac{Q}{2\pi rL}.

Given:

  • Q=10 μC=10×10−6 CQ=10\,\mu C =10\times10^{-6}\,C
  • L=1 km=1000 mL=1\,km=1000\,m

If the radius is assumed (or given in the diagram) to be r=10−5 mr=10^{-5}\,m, then

σ=10×10−62π (10−5)(1000)=10×10−62π×10−2=10×10−60.0628≈1.59×10−4 C/m2.\sigma=\frac{10\times10^{-6}}{2\pi\, (10^{-5})(1000)} =\frac{10\times10^{-6}}{2\pi\times10^{-2}} =\frac{10\times10^{-6}}{0.0628} \approx 1.59\times10^{-4}\,C/m^2.