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Fernando Garcia

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PHYS405 - Electricity & Magnetism I

Spring 2024

Problem 1.53

To gain geometrical intuition as to when a vector field is the gradient of scalar or the curl of another vector field, it is relevant to see what these fields look like: (In order ${\mathbf{v}}_a,{\mathbf{v}}_b,{\mathbf{v}}_c$)

a) Which of them can be expressed as the gradient of a scalar? Find a scalar function that does the job.

Recall that a vector field has a scalar potential if and only if the vector field is curl-less (also known as an irrotational field):

$$\begin{array}{}\text{(Eq. 1.103, page 51)}& \mathrm{\nabla }\times \mathbf{F}=\mathbf{0}\iff \mathbf{F}=-\mathrm{\nabla }V\end{array}$$

By recalling that the curl is given by:

$$\begin{array}{}\text{(As seen on the cover of your book)}& \mathrm{\nabla }\times \mathbf{v}=(\frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z})\hat{\mathbf{x}}+(\frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x})\hat{\mathbf{y}}+(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y})\hat{\mathbf{z}}\end{array}$$

It is straightforward to see that

$$\begin{array}{rl}\mathrm{\nabla }\times {\mathbf{v}}_a& =-6xz\hat{\mathbf{x}}+2z\hat{\mathbf{y}}+3z^2\hat{\mathbf{z}}\\ \mathrm{\nabla }\times {\mathbf{v}}_b& =-2y\hat{\mathbf{x}}-3z\hat{\mathbf{y}}-x\hat{\mathbf{z}}\\ \mathrm{\nabla }\times {\mathbf{v}}_c& =\mathbf{0}\end{array}$$

Thus, ${\mathbf{v}}_c$ is the only vector field out of those 3 which can be written as the gradient of a scalar field.

b) Which can be expressed as the curl of a vector? Find such a vector.

Recall that a vector field has a vector potential if and only if the vector field is divergence-less (also known as an solenoidal field):

$$\begin{array}{}\text{(Eq. 1.104, page 52)}& \mathrm{\nabla }\cdot \mathbf{F}=0\iff \mathbf{F}=\mathrm{\nabla }\times \mathbf{A}\end{array}$$

By recalling that the divergence is given by:

$$\begin{array}{}\text{(As seen on the cover of your book)}& \mathrm{\nabla }\cdot \mathbf{v}=\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}\end{array}$$

It is straightforward to see that

$$\begin{array}{rl}\mathrm{\nabla }\cdot {\mathbf{v}}_a& =0\\ \mathrm{\nabla }\cdot {\mathbf{v}}_b& =3x+y+2z\\ \mathrm{\nabla }\cdot {\mathbf{v}}_c& =2x+2y\end{array}$$

Thus, ${\mathbf{v}}_a$ is the only vector field out of those 3 which can be written as the gradient of a scalar field.

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