(a) Show that ∇·(𝐞r / r^2)=0 for r>0 . Hint: You can do this in rectangular coordinates, but it

step 1 For a radial vector field equals f of r e sub r, spherical coordinates. So this is going to depe

(a) Show that ∇·(𝐞r / r^2)=0 for r>0 . Hint: You can do this...

(a) Show that $\nabla \cdot\left(\mathbf{e}_{r} / r^{2}\right)=0$ for $r>0 .$ Hint: You can do this in rectangular coordinates, but it is easier in spherical coordinates. See Chapter 6, equation (7.9). Show that $\nabla \cdot\left[\mathbf{e}_{r} F(r)\right],$ for any $F(r),$ is a function of $r$ only. (b) Show that $\nabla(1 / r)=-\mathbf{e}_{r} / r^{2} .$ See Chapter $6,$ equation (6.8)

(a) Show that $\nabla \cdot\left(\mathbf{e}_{r} / r^{2}\right)=0$ for $r>0 .$ Hint: You can do this in rectangular coordinates, but it is easier in spherical coordinates. See Chapter 6, equation (7.9). Show that $\nabla \cdot\left[\mathbf{e}_{r} F(r)\right],$ for any $F(r),$ is a function of $r$ only.
(b) Show that $\nabla(1 / r)=-\mathbf{e}_{r} / r^{2} .$ See Chapter $6,$ equation (6.8)

Key Concepts

Divergence Operator

The divergence operator measures the net rate of flux expansion or contraction at a point in a vector field. It is a scalar operator that helps determine whether a point is a source or a sink. In problems involving spherical symmetry, calculating the divergence in spherical coordinates can simplify computations, as the radial dependence is separated from angular dependencies.

Spherical Coordinate System

Spherical coordinates are a three-dimensional coordinate system where points are specified by a radius (r) and two angles. This coordinate system is particularly useful for problems with radial symmetry, such as those involving fields that depend only on the distance from the origin. In these coordinates, both the divergence and gradient operators have forms that explicitly isolate the radial component, simplifying calculations for radially dependent functions.

Radial Vector Fields

Radial vector fields are fields where the vectors point directly away from or toward the origin, typically represented as a scalar function multiplied by the unit radial vector. When the scalar function depends solely on the radius, the overall field inherits spherical symmetry, and its divergence becomes a function solely of r. Analyzing the divergence of such fields often leads to simplified expressions due to the cancellation of angular dependencies.

Gradient Operator

The gradient operator acts on scalar fields to produce vector fields that indicate the direction and rate of greatest increase. In the context of potentials, such as the 1/r potential, the gradient determines the associated field (often called the force field in physics). The result, for example, that the gradient of 1/r equals -e_r/r^2, illustrates how the potential function changes with distance and why the direction points radially, aligning with the structure of radially symmetric fields.

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