We have an initially uncharged hollow metallic sphere with radius of 5.0 cm. I place a small object with a charge of +20 µC at the center of the sphere through a hole in the surface. Find the electric field present at a point 10 cm from the sphere’s center. (ke = 8.99 × 109 N⋅m2/C2)

a. 3.6E+7 N/C
b. 9.0E+6 N/C
c. 2.7E+7 N/C
d. 1.8E+7 N/C
e. 4.5E+6 N/C

d. 1.8E+7 N/C

Explanation:

In order to find the electric field outside the hollow metallic sphere, we can apply Gauss' Law, using a spherical gaussian surface with radius equal to 10 cm from the center of the sphere.

As the electric field must be normal to the surface at any point (no tangential fields can exist in electrostatic conditions) it must be radially pointed. By symmetry, at a same radius, the magnitude of the field must be the same.

As the dA vector is always normal to the surface and aiming outward, the dot product E*dA, can be taken out of the integral, as follows:

where Qenc, is the total charge enclosed by the gaussian surface. Just due to the conservation of charge, this charge must be equal to +20 μC.

Now, how can this charge be distributed on the outer surface of the shell?

If we apply Gauss´Law to a gaussian surface with a radius just inside the sphere (between the inner and outer surface), we will find that the flux is 0, due to the electric field is 0 inside a conductor.

We could write the same equation as above:

If the left side of the equation is 0, the right one must be zero too:

⇒ Qenc = 0  ⇒ Qc + Qin = 0 ⇒ Qin = -20 μC.

As the metallic sphere must remain neutral, an equal and opposite charge must build up on the outer surface:

⇒ Qou = +20 μC

The other parameters in the equation are:

r = 0.1 m

ε₀ = 8.85*10⁻¹² C²/N*m²

Replacing by the values, we can solve for E as follows:

⇒ E = 1.8*10⁷ N/C

So, the statement d. is true.

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