A solid metallic sphere of radius a carries total charge Q ....

A solid metallic sphere of radius $a$ carries total charge $Q$ . No other charges are nearby. The electric field just outside its surface is $k_{e} Q / a^{2}$ radially outward. At this close point, the uniformly charged surface of the sphere looks exactly like a uniform flat sheet of charge. Is the electric field here given by $\sigma / \epsilon_{0}$ or by $\sigma / 2 \epsilon_{0} ?$

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the net electric flux through a closed surface to the charge enclosed by that surface. In problems involving symmetric charge distributions, Gauss's Law allows for a straightforward derivation of the electric field. For this problem, applying Gauss's law to a small Gaussian surface straddling the charged metallic surface shows that the discontinuity in the electric field is directly proportional to the surface charge density.

Electric Field at a Conducting Surface

The electric field just outside a conductor in electrostatic equilibrium has a magnitude of ?/??, where ? is the surface charge density. This result emerges because the field inside the conductor is zero and the entire discontinuity in the electric field occurs at the surface. This contrasts with the electric field due to an infinite sheet of charge, which is ?/(2??) on each side of the sheet, highlighting that on the surface of a conductor, the field is entirely on one side.

Surface Charge Density

Surface charge density, denoted by ?, is defined as the amount of charge per unit area on a surface. It is a key parameter when analyzing the electric fields generated by charged objects. In the context of this problem, ? is related to the total charge Q on the sphere and its surface area, ultimately affecting the magnitude of the electric field just outside the sphere as derived using Gauss's Law.

Comparison with Infinite Sheet of Charge

An infinite sheet of charge produces an electric field of ?/(2??) on either side because the field is symmetrically distributed in both directions. However, when a conductor is involved, the electric field exists only outside the conductor while inside the field is zero. Hence, near the surface of a conductor the effective electric field becomes ?/??, revealing that local flatness does not imply an infinite sheet behavior in terms of field calculations.

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