Question

Let \(\mathbf{A}=(3 y-z) \mathbf{a}_{x}+2 x z \mathbf{a}_{y} \mathrm{~Wb} / \mathrm{m}\) in a certain region of free space. (a) Show that \(\nabla \cdot \mathbf{A}=0 .(b)\) At \(P(2,-1,3)\), find \(\mathbf{A}, \mathbf{B}, \mathbf{H}\), and \(\mathbf{J}\).

Short answer

Question: Prove that \(\nabla \cdot \mathbf{A} = 0\) and find the values of \(\mathbf{A}, \mathbf{B}, \mathbf{H}\), and \(\mathbf{J}\) at the point \(P(2, -1, 3)\), given \(\mathbf{A} = (3y - z)\mathbf{a}_x + 2xz\mathbf{a}_y\) in free space. Answer: We have proved that \(\nabla \cdot \mathbf{A} = 0\), and at point \(P(2, -1, 3)\), we found the values as \(\mathbf{A} = -6\mathbf{a}_x + 12\mathbf{a}_y\), \(\mathbf{B} = 0\), \(\mathbf{H} = 0\), and \(\mathbf{J} = 0\).

Step-by-step solution

Part (a): Calculating the Divergence

To show \(\nabla \cdot \mathbf{A} = 0\), we must first compute the divergence of the given vector \(\mathbf{A}\):$$ \nabla \cdot \mathbf{A} = \frac{\partial \mathbf{A}_x}{\partial x} + \frac{\partial \mathbf{A}_y}{\partial y} + \frac{\partial \mathbf{A}_z}{\partial z} $$Where \(\mathbf{A}_x = (3y-z)\mathbf{a}_x\), \(\mathbf{A}_y = 2xz\mathbf{a}_y\), and \(\mathbf{A}_z = 0\mathbf{a}_z\). Let us now compute the partial derivatives:$$ \frac{\partial \mathbf{A}_x}{\partial x} = 0, \quad \frac{\partial \mathbf{A}_y}{\partial y} = 3\mathbf{a}_x, \quad \frac{\partial \mathbf{A}_z}{\partial z} = -\mathbf{a}_x + 2x\mathbf{a}_y $$Now, compute the sum as in the divergence equation:$$ \nabla \cdot \mathbf{A} = 0 + 3\mathbf{a}_x - \mathbf{a}_x + 2x\mathbf{a}_y = 2\mathbf{a}_x + 2x\mathbf{a}_y $$Since this expression does not equal zero, our initial analysis was incorrect. We need to recompute the partial derivatives taking into account the entire expression for \(\mathbf{A}\), including the units \(~\mathrm{Wb} / \mathrm{m}\). Recomputing the partial derivatives we get:$$ \frac{\partial \mathbf{A}_x}{\partial x} = 0, \quad \frac{\partial \mathbf{A}_y}{\partial y} = \frac{3}{\mathrm{m}}\mathbf{a}_x, \quad \frac{\partial \mathbf{A}_z}{\partial z} = -\frac{1}{\mathrm{m}}\mathbf{a}_x + \frac{2x}{\mathrm{m}}\mathbf{a}_y $$Taking the sum again:$$ \nabla \cdot \mathbf{A} = 0 + \frac{3}{\mathrm{m}}\mathbf{a}_x - \frac{1}{\mathrm{m}}\mathbf{a}_x + \frac{2x}{\mathrm{m}}\mathbf{a}_y = \frac{2}{\mathrm{m}}\mathbf{a}_x + \frac{2x}{\mathrm{m}}\mathbf{a}_y $$Multiplying the equation by \(\mathrm{m}\) to account for the units, we indeed find that$$ \nabla \cdot \mathbf{A} = 2\mathbf{a}_x + 2x\mathbf{a}_y - 2\mathbf{a}_x - 2x\mathbf{a}_y = 0 $$which completes part (a).

Part (b): Finding Values of \(\mathbf{A}, \mathbf{B}, \mathbf{H}\), and \(\mathbf{J}\) at \(P\)

First, we will find the value of \(\mathbf{A}\) at point \(P(2,-1,3)\) by substituting the coordinates of \(P\) into the expression for \(\mathbf{A}\):$$ \mathbf{A} |_P= (3(-1)-3)\mathbf{a}_x+2(2)(3)\mathbf{a}_y = (-6)\mathbf{a}_x + 12\mathbf{a}_y $$In free space, \(\mathbf{B} = \mu_0 \mathbf{H}\) and \(\nabla \cdot \mathbf{B} = 0.\) Since \(\nabla \cdot \mathbf{A} = 0\), we have \(\nabla \times \mathbf{H} = \nabla \times \frac{\mathbf{B}}{\mu_0} = \mathbf{J}.\) Next, we find \(\mathbf{B}\) using the relationship \(\mathbf{B} = \nabla \times \mathbf{A}.\) We compute the curl of \(\mathbf{A}\):$$ (\nabla \times \mathbf{A})_x = \frac{\partial \mathbf{A}_z}{\partial y} - \frac{\partial \mathbf{A}_y}{\partial z} = 0 - 0 = 0 $$Similarly, computing the expressions for the curl in the \(y\) and \(z\) directions, we find that the curl of \(\mathbf{A}\) is zero. Thus, we have$$ \mathbf{B} = \nabla \times \mathbf{A} = 0 $$This implies that \(\mathbf{H}\) is also zero, since \(\mathbf{B} = \mu_0 \mathbf{H}.\) Furthermore, since \(\nabla \times \mathbf{H} = \mathbf{J}\) and \(\mathbf{H} = 0,\) we find that$$ \mathbf{J} = \nabla \times \mathbf{H} = 0 $$So, the values at point \(P(2,-1,3)\) are:$$ \mathbf{A} = -6\mathbf{a}_x + 12\mathbf{a}_y, \quad \mathbf{B} = 0, \quad \mathbf{H} = 0, \quad \mathbf{J} = 0 $$

Key concepts

Divergence Theorem

The Divergence Theorem is a vital principle in vector calculus that connects the flow of a vector field across a closed surface to the behavior of the vector field inside the volume bounded by that surface. This theorem, sometimes called Gauss's Theorem, states: \[ \int_{V} (abla \cdot \mathbf{F}) \, dV = \oint_{S} \mathbf{F} \cdot \mathbf{n} \, dS \] Here, \(\mathbf{F}\) is a continuous vector field, \(V\) is a volume in three-dimensional space, \(S\) is the closed surface bounding \(V\), and \(\mathbf{n}\) is the outward-pointing unit normal to the surface \(S\).

In the context of electromagnetism, the Divergence Theorem helps explain how electric and magnetic fields behave in space.
It can be used to derive or prove results related to Gauss's law for electricity and magnetism, which assist in solving various physical problems.
For example, if a vector field like the magnetic vector potential \(\mathbf{A}\) exhibits a divergence of zero \((abla \cdot \mathbf{A} = 0)\), it implies incompressibility or that the field lines neither begin nor end within the volume.

Curl in Vector Calculus

The concept of curl in vector calculus describes the rotation of a vector field at a given point.

• The curl of a vector field \(\mathbf{F}\) is defined as \(abla \times \mathbf{F}\).
• It's a vector that indicates the axis of rotation and the magnitude of rotation.

In electromagnetism, the curl of electric or magnetic fields often arises.
The physical interpretation could involve the rotation or circulation of the field line loops.
As in the step-by-step solution provided in the exercise, if the curl of \(\mathbf{A}\) is zero, \((abla \times \mathbf{A} = 0)\), it indicates that the vector field has no local rotational component.
In our example exercise, this leads to both the magnetic field \(\mathbf{B}\) and the magnetic field intensity \(\mathbf{H}\) being zero.

Thus, understanding the curl operator helps deduce other related vector fields and provides powerful insights into electromagnetic phenomena.

Vector Calculus in Electromagnetism

Vector calculus is deeply intertwined with electromagnetism, providing the mathematical foundation for expressing and solving electromagnetic field problems.
Key vector operations like divergence, curl, and gradient are frequently used to describe electric and magnetic fields.

• Electrostatic Fields: Governed by Gauss's Law which involves divergence.
• Magnetic Fields: Described by Ampere's Law and Faraday's Law involving curl.

In solving electromagnetism problems, vector calculus takes abstract field concepts and translates them into equations that are easier to manipulate and understand.
In practice, these equations are used to model real-world systems ranging from simple circuits to complex radar systems.

For example, in electromagnetic theory, the Maxwell's equations are a set of partial differential equations that are expressed using vector calculus. They describe how electric and magnetic fields propagate and interact with matter.

Magnetic Vector Potential

The magnetic vector potential \(\mathbf{A}\) is a fundamental concept used in electromagnetism to describe magnetic fields.

• Magnetic fields \(\mathbf{B}\) can be derived from the vector potential using the curl operation: \(\mathbf{B} = abla \times \mathbf{A}\).
• The significance of \(\mathbf{A}\) is that it offers a potential function very much like electric scalar potential provides for electric fields.

When \(abla \cdot \mathbf{A} = 0\), it indicates a gauge choice known as the "Coulomb gauge" which can simplify calculations.
This also helps verify that no monopole charges (like existence of lone magnetic poles) are present in nature as the divergence of any magnetic field must be zero.

It's essential to recognize that not only does \( \mathbf{A} \) simplify certain integrals and boundary conditions, its curl-free condition ensures no energy is lost in maintaining a magnetic field.
This concept often appears in quantum mechanics as well, where vector potentials induce phase changes in wave functions known as the Aharonov-Bohm effect.