Are the following possible as specifications of B-fields? (a) \(B_{x}=k x, B_{y}=k y, B_{z}=k z\). (b) \(B_{x}=k x, B_{y}=0, B_{x}=-k a\). (c) \(B_{x}=k\left(x^{2}+y^{2}\right), B_{y}=-k\left(x^{2}+y^{2}\right), B_{s}=0\).

When discussing magnetic field specifications, we are referring to the characteristics that define a magnetic field in a given region of space. These can be represented mathematically by field components, such as \( B_x \) for the x-direction, \( B_y \) for the y-direction, and \( B_z \) for the z-direction.

A correct magnetic field specification must align with the fundamental principles of electromagnetism, one of which is the divergence of the magnetic field being zero. This is represented by \( abla \cdot \mathbf{B} = 0 \), indicating that magnetic monopoles do not exist and that magnetic field lines are continuous loops.

For example, a linear specification \( B_x = kx, B_y = ky, B_z = kz \) implies that the further away from the origin, the stronger the magnetic field. However, as shown in the exercise solution, such a field would not have zero divergence and therefore cannot represent a physical magnetic field.

Gauss's law for magnetism is one of the four equations of Maxwell's equations. It states that the magnetic flux through a closed surface is zero, which mathematically means that the divergence of the magnetic field \( \mathbf{B} \) is always zero: \( abla \cdot \mathbf{B} = 0 \).

What this law implies is that unlike electric fields, where flux can be nonzero due to electric charges, magnetic poles always come in north and south pairs, so-called dipoles. Therefore, magnetic lines enter and exit a closed surface an equal number of times, resulting in net zero flux. This is why none of the field specifications in parts (a) or (b) of the textbook exercise fulfill Gauss's law for magnetism, since their divergence is not zero everywhere.

Partial derivatives play a crucial role in electromagnetism, especially when calculating the divergence of a field, as seen in the exercise. They allow us to determine how a field changes with respect to one variable while holding the other variables constant. For a three-dimensional magnetic field \( \mathbf{B} \), the divergence involves the partial derivatives of each component field with respect to its corresponding direction: \( \frac{\partial B_x}{\partial x} \) for x, \( \frac{\partial B_y}{\partial y} \) for y, and \( \frac{\partial B_z}{\partial z} \) for z.

Following Gauss's law for magnetism, the sum of these partial derivatives must be zero for the magnetic field to be physically plausible. In the evaluated example for (c), the divergence is not constant and can be zero, demonstrating the importance of understanding and using partial derivatives to verify the legitimacy of magnetic field specifications.