2 Electromagnetism
2.1 EXPLANATIONS
2.1.1 Maxwell’s Equations
Maxwell’s equations express, formally, the properties of electromagnetic waves.
Gauss’s Law for Electricity \[ \nabla \cdot \mathbf{D} = \rho_v \]
Gauss’s Law for Magnetism \[ \nabla \cdot \mathbf{B} = 0 \]
Faraday’s Law \[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
Ampere-Maxwell Law \[ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} \]
Where
- \(\nabla\) = Vector differential operator.
- \(\mathbf{D}\) = Electric flux density.
- \(\mathbf{B}\) = Magnetic flux density.
- \(\mathbf{E}\) = Electric field intensity.
- \(\mathbf{H}\) = Magnetic field intensity.
- \(\rho_v\) = Volume charge density.
- \(\mathbf{J}\) = Current density.
- \(\frac{\partial \mathbf{D}}{\partial t}\) = Displacement current density.
2.1.2 Stoke’s Theorem Physics
Stokes’ theorem relates the circulation of a vector field \(\mathbf{A}\) around a closed path \(L\) to the surface integral of its curl over an open surface \(S\) bounded by \(L\) [1]:
\[ \int_S (\nabla \times \mathbf{A}) \cdot d\mathbf{S} = \oint_L \mathbf{A} \cdot d\mathbf{l}\]
Where: * \(d\mathbf{S}\) is the surface element vector [1]. * The orientation follows the right-hand rule [2].
2.1.3 Stoke’s Theorem for Flux Integral
\[\iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dA = \oint_C \mathbf{F} \cdot \mathbf{r}'(s) \, ds\].
Or,
\[\iint_S (\text{curl} \times \mathbf{F}) \cdot \mathbf{n} \, dA = \oint_C \mathbf{F} \cdot \mathbf{r}'(s) \, ds\].
In this formula: * \(S\) a piecewise smooth oriented surface in space * \(C\) a piecewise smooth simple closed curve boundary of \(S\) * \(\mathbf{n}\) is the unit normal vector of \(S\). * \(\mathbf{r}'(s)\) is the unit tangent vector of \(C\). * \(s\) is the arc length of \(C\). * \(\mathbf{F}(x, y, z)\) is a continuous vector function
2.1.4 Surface Integrals and Fundamental Theorem of Calculus
\[\int_D d\omega = \int_{\partial D} \omega\]
Where
- \(\omega\): A differential \((k-1)\)-form.
- \(d\omega\): The exterior derivative of the form \(\omega\).
- \(D\): A \(k\)-dimensional domain of integration.
- \(\partial D\): The oriented boundary of the domain \(D\).
2.2 GLOSSARY
Principle of Superposition
See Physics Mathemematical Appendix.
Electromagnetism The phenomena of interactions between charges at rest and in motion.
Electromagnetic Waves Oscillations of electric and magnetic fields that propagate through space without the need for charges or currents.
Reference Frame A coordinate system where measurements can be made about the time and position about objects.
Inertial Reference Frame A reference frame based on Newton’s First Law, where all objects are at rest or in motion unless influenced by external force.
Phase
Gradient
\(\nabla V\) Operating on a scalar function \(V\) produces a vector field.
Divergence
\(\nabla \cdot \mathbf{A}\) The dot product of the operator with a vector field \(\mathbf{A}\) yields a scalar field.
Curl
\(\nabla \times \mathbf{A}\) The cross product of the operator with a vector field \(\mathbf{A}\) results in another vector field.
Laplacian
\(\nabla^2 V\) This operator represents the divergence of the gradient (\(\nabla \cdot \nabla V\)).
2.3 MATHEMARICS APPENDIX
Stoke’s Theorem
Let \(S\) be a piecewise smooth oriented surface in space, and let the boundary of \(S\) be a piecewise smooth simple closed curve \(C\). If with continuous first partial derivatives in a domain containing \(S\), then:
\[\iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dA = \oint_C \mathbf{F} \cdot \mathbf{r}'(s) \, ds\].
Or,
\[\iint_S (\text{curl} \times \mathbf{F}) \cdot \mathbf{n} \, dA = \oint_C \mathbf{F} \cdot \mathbf{r}'(s) \, ds\].
In this formula: * \(\mathbf{n}\) is the unit normal vector of \(S\). * \(\mathbf{r}'(s)\) is the unit tangent vector of \(C\). * \(s\) is the arc length of \(C\). * \(\mathbf{F}(x, y, z)\) is a continuous vector function
Flux
Total amount of a field or fluid passing through a surface.
Double Integral
The double integral is defined by the following identity: \[\iint_R f(x, y) \, dA = \lim_{m, n \to \infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*, y_{ij}^*) \Delta A\].
where * \(R\) is a closed bounded region in the \(xy\)-plane over which the function is integrated [1, 2]. * \(m\) and \(n\) are the number of subrectangles into which the region \(R\) is partitioned [3, 4]. * \(R_{ij}\) are the smaller subrectangles resulting from dividing \(R\) using a grid of lines parallel to the axes [4, 5]. * \(\Delta A\) represents the area of each subrectangle \(R_{ij}\) [4, 6]. * \((x_{ij}^*, y_{ij}^*)\) is an arbitrary sample point chosen within the subrectangle \(R_{ij}\) [4, 6]. * The limit is taken as \(m\) and \(n\) approach infinity such that the maximum diameter (the greatest distance between any two points within a subregion) approaches zero [3, 7]. * The function \(f(x, y)\), the integrand, is typically assumed to be continuous on \(R\) to ensure the limit exists and is independent of the choice of sample points [7, 8].
Double Integral Notational Variations
There exist double integral with two equivalent notations: * Differential Area Form: \(\displaystyle \iint_R f(x, y) \, dA\). * Coordinate Form: \(\displaystyle \iint_R f(x, y) \, dx \, dy\).
Vector Differential Operator
The vector differential operator for expressing the partial differentiation of a vector
\[\nabla = \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{k}\]
where
\(\nabla\) is the vector differential operator, commonly referred to as del or nabla.
\(\dfrac{\partial}{\partial x}, \dfrac{\partial}{\partial y}, \text{ and } \dfrac{\partial}{\partial z}\) are partial derivatives.
\(\mathbf{i}, \mathbf{j}, \text{ and } \mathbf{k}\) are the standard basis unit vectors pointing in the positive directions of the \(x, y, \text{ and } z\) axes, respectively
Curl
Let \(\mathbf{v}(x, y, z)\) be a differentiable vector function expressed in Cartesian coordinates with components \([v_1, v_2, v_3]\), such that: \[\mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k}\] The curl of \(\mathbf{v}\) (also denoted as \(\nabla \times \mathbf{v}\) or \(\text{rot } \mathbf{v}\)) is defined by the following symbolic determinant: \[\text{curl } \mathbf{v} = \nabla \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ v_1 & v_2 & v_3 \end{vmatrix}\]
Divergence
Differentiation of Vector
\[\mathbf{v}'(t) = \lim_{\Delta t \to 0} \frac{\mathbf{v}(t + \Delta t) - \mathbf{v}(t)}{\Delta t}\].
Vector Function
\[\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}\]
where - \(f, g, h\) are components of the \(\mathbf{r}(t)\)
2.4 PHYSICS MATHEMATICAL APPENDIX
Principle of Superposition
\[D_\text{net} = D_{1}+D_{2}+\cdots=\sum_{i}D\]
where - \(D_{i}\) is the displacement caused by each wave