Using Gauss's law, derive an expression for the electric field intensity at any point outside a uniformly charged thin spherical shell of radius R and charge density a C/m². Draw the field lines when the charge density of the sphere is (i) positive, (ii) negative. (b) A uniformly charged conducting sphere of 2.5 m in diameter has a surface charge density of 100 μC/m². Calculate the
(i) charge on the sphere
(ii) total electric flux passing through the sphere.

Solution

(a) Electric field intensity at any point outside a uniformly charged spherical shell:

Consider a thin spherical shell of radius R with centre O. Let charge +q is uniformly distributed over the surface of the shell.

Let P be any point on the sphere S₁ with centre O and radius r. According to Gauss's Law

If σ is the surface charge density,

∴ $q = σ . A = σ . 4 π r^{2}$

∴ Electric field lines when the charged density of the sphere:

(i) Positive (ii) Negative

(b)
Here given,
diameterof charged conducting sphere = 2.5 m
$∴ Radius - r = \frac{2.5}{2} = 1.25 m$
Charge density is given by $σ = 100 μc / m^{2} = 100 \times 10^{- 6} = 10^{- 4} C / m^{2}$

(i)charge on the sphere - $q = 4 π R^{2} σ$
$= 4 \times 3.14 \times (1.25)^{2} \times 10^{- 4} = 19.625 \times 10^{- 4} = 1.96 \times 10^{- 3} C$
(ii) Total electric flux
$ϕ_{E} = \frac{q}{\in_{0}}$
$∴$ $ϕ_{E} = \frac{1.96 \times 10^{- 3}}{8.85 \times 10^{- 12}} = 0.221 \times 10^{9} = 2.21 \times 10^{8} {Nm}^{2} C^{- 1}$

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Electric Charges And Fields

Some More Questions From Electric Charges and Fields Chapter

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
b) Explain why two field lines never cross each other at any point?

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