Question

An uncharged, nonconducting, hollow sphere of radius 10.0cm surrounds a 10.0-μC charge located at the origin of a Cartesian coordinate system. A drill with a radius of 1.00mm is aligned along the z axis, and a hole is drilled in the sphere. Calculate the electric flux through the hole.

Answer 1

The electric flux through the hole is $56.45\ webber$ .

• Electric flux is the number of electric field lines cutting through the surface and is measured as surface intregal of electric field over that surface
• Mathematically it is given by $\phi_E=E.A \ Nm^2/C$ where E is the electric field and A is the area.
• Gauss's law states that electric flux through closed surface is equal to the 1 / ε₀ times the charge enclosed by that surface which is given by  Ф = q / ε₀ where q is the central charge and ε₀ is the permittivity of the medium.

It is given , hollow sphere of radius 10.0cm surrounds a 10.0-μC charge.

The whole surface of hollow sphere $= 4\pi r^2$

                                                            $= 4\times 3.14\times (10 \times 10^{-2})^2 \\\\= 12.56\times 10^{-2} m^2$

Area of the hole ( both side ) $= 2\times \pi r^2$

                                               $= 2\times 3.14 \times (10^-^3)^2\\= 6.28 \times 10^-^6 m^2$

According to Gauss's theorem, the flow from a particular charge in the center is given by

 $\phi= \frac{10\times10^-^6}{8.85\times 10^-^1^2}\\\\\phi=1.13\times10^6$

This flux flows through the surface of the sphere, so the flux  per unit area which is given by

$= \frac{ 1.13\times 10^6 }{ 12.56\times 10^-^2} \\\\= 8.99 \times 10^6 \ weber / m^2$

Flux through area of hole is given by :

$= 8.99\times10^6 \times6.28 \times 10^-^6\\ = 56.45 \ weber$

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