A hollow conducting sphere of radius R carries a negative...

A hollow conducting sphere of radius $R$ carries a negative charge $-q .$ (a) Write expressions for the electric field $\overrightarrow{\mathbf{E}}$ inside $(r<R)$ and outside $(r>R)$ the sphere. Also indicate the direction of the field. (b) Sketch a graph of the field strength as a function of $r$. [Hint: See Conceptual Example 16.7.]

Key Concepts

Spherical Charge Distributions

A charged spherical object, especially one with uniformly distributed surface charge, produces an electric field outside the sphere that can be treated as if all of its charge were concentrated at its center. This effectively reduces the problem to finding the field of a point charge, with the field magnitude varying inversely with the square of the distance from the center.

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux through a closed surface to the charge enclosed by that surface. In problems with high symmetry, such as a spherical charge distribution, it allows one to calculate the electric field by considering an appropriately chosen Gaussian surface, greatly simplifying the calculation.

Electric Field in Conductors

In electrostatics, the electric field inside a conductor in equilibrium is zero. This is because free charges within the conductor move until they cancel any internal fields. For hollow conductors, this implies that the electric field at any point inside the conducting material (and within any cavities that do not enclose additional charges) is zero.

Graphical Representation of Electric Field

When graphing the electric field strength as a function of distance from a sphere, the graph typically shows a zero field inside a conductor, and a field that decreases as 1/r^2 outside the conductor. This graph helps illustrate how the field transitions from zero within the conductor to a diminishing value at larger distances, emphasizing the role of symmetry and the inverse-square nature of the field.

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