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The electric field at point P, just outside the outer surface of a hollow spherical conductor with an inner radius of 10 cm and an outer radius of 20 cm, has a magnitude of 459 N/C and is directed outward. When an unknown point charge \( Q \) is introduced into the center of the sphere, the electric field at P is still directed outward but is now 181 N/C.

(a) What is the net charge on and enclosed by the outer spherical surface before \( Q \) is introduced?

Answer :

Answer:

[tex]2.042\times 10^{-9}\ C.[/tex]

Explanation:

Given:

  • Electric field just outside the given sphere, E = 459 N/C.
  • Inner radius of the shell, [tex]\rm r_i[/tex] = 10 cm = 0.1 m.
  • Outer radius of the shell, [tex]\rm r_o[/tex] = 20 cm = 0.2 m.
  • Electric field outside the shell after introducing the charge Q, [tex]\rm E_f[/tex] = 181 N/C.

Before introducing the charge Q,

Consider a Gaussian sphere of same radius as the outer radius of the shell, concentric with the given shell,

Applying Gauss' law over this surface,

[tex]\oint \vec E \cdot d\vec A = \rm \dfrac{q}{\epsilon_o}.[/tex]

where,

  • [tex]\epsilon_o[/tex] is the electrical permittivity of free space having value [tex]9\times 10^9\rm \ Nm^2/C^2[/tex].
  • [tex]d\vec A[/tex] is the surface area element of the sphere, directed along the normal to the plane of the surface.
  • [tex]\rm q[/tex] is the net charge enclosed by the Gaussian surface.

The directions of the area element and the electric field, both are directed outward the surface, therefore, [tex]\vec E \cdot d\vec A = E\ dA[/tex].

The LHS of the equation of the Gauss' law is given as

[tex]\oint \vec E\cdot d\vec A=\oint E\ dA =E\oint dA[/tex]

[tex]\oint dA[/tex] is the surface area of the Gaussian sphere = [tex]\rm 4\pi r_f^2[/tex]

Therefore, using the equation of the Gauss' law,

[tex]\oint \vec E\cdot d\vec A=\rm E\ 4\pi r_f^2=\dfrac{q}{\epsilon_o}\\\Rightarrow q =E\ 4\pi r_f^2\times \epsilon_o\\=459\times 4\pi \times (0.2)^2\times 8.85\times 10^{-12}\\=2.042\times 10^{-9}\ C.[/tex]

It is the net charge enclosed by the outer surface before Q was introduced.

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