Introduction to Electromagnetism

The divergence of A is represented as $ abla imes A = rac{rac{ ext{ extd}A_x}{ ext{ extd}x} + rac{ ext{ extd}A_y}{ ext{ extd}y} + rac{ ext{ extd}A_z}{ ext{ extd}z}}$.

A positive divergence indicates a source where the vector field A is emanating from a point.

Zero divergence means that the field has neither a source nor a sink, indicating that the flux entering an element equals the flux leaving it.

<p>Curl provides a measure of the rotation or swirling of a vector field around a point, represented as the cross product of the del operator and the vector.</p> Signup and view all the answers

<p>A solenoidal vector function is one where the divergence is zero everywhere, implying constant flux across any closed surface.</p> Signup and view all the answers

<p>The current flowing through a capacitor is given by $I_C = C \frac{dV}{dt}$.</p> Signup and view all the answers

<p>Displacement current $I_d$ occurs when the voltage across a capacitor is changing, allowing current to flow even in the absence of conduction current.</p> Signup and view all the answers

<p>Displacement current density $J_d$ is defined as $J_d = \frac{dD}{dt}$, where $D$ is the electric displacement field.</p> Signup and view all the answers

<p>Permittivity $\epsilon$ relates the electric field intensity to the electric displacement field, defined as $D = \epsilon E$.</p> Signup and view all the answers

<p>Maxwell's equations describe how electric and magnetic fields interact, foundational to understanding electromagnetic phenomena.</p> Signup and view all the answers

<p>The electric field intensity $E$ across a capacitor is given by $E = \frac{V}{d}$, where $d$ is the separation between the plates.</p> Signup and view all the answers

<p>This equation represents the displacement current $I_d$ in terms of permittivity, area, plate spacing, and the rate of change of electric field.</p> Signup and view all the answers

<p>In the absence of conduction current, Ampere's circuital law reduces to $ abla \times \vec{H} = \oint_S \vec{J_d}.ds$.</p> Signup and view all the answers

<p>The divergence of the electric field is given by $div E = \frac{P}{\epsilon_0}$, indicating that the electric field's behavior is directly related to the volume charge density.</p> Signup and view all the answers

<p>The curl of the electric field is given by $\nabla \times E = 0$, indicating that in electrostatic conditions, the electric field is conservative and has no circulation around any closed path.</p> Signup and view all the answers

<p>Poisson's equation is $\nabla^2 \phi = -\frac{P}{\epsilon_0}$, and it relates the electric potential to the charge density in a region, providing a link between electrostatics and potential theory.</p> Signup and view all the answers

<p>Laplace's equation is $\nabla^2 \phi = 0$, which indicates that in regions with no charge present (i.e., $P=0$), the electric potential satisfies this equation.</p> Signup and view all the answers

<p>The integral form, $\oint_S E.ds = \frac{Q}{\epsilon_0}$, reflects the total electric flux through a closed surface being proportional to the charge enclosed, defining how electric fields emanate from charges.</p> Signup and view all the answers

<p>The potential difference between two points is computed using $V_{AB} = - \int_{B}^{A} E.dl$, demonstrating that the potential depends only on the endpoints, not the path taken.</p> Signup and view all the answers

<p>$\oint E.dl = 0$ signifies that the work done around any closed path in a static electric field is zero, implying that the electric field is conservative.</p> Signup and view all the answers

<p>Applying Gauss's Divergence theorem allows us to write the flux of the electric field through a surface in terms of the volume charge density, establishing $\oint_S E.ds = \int_V (\nabla.E)dv$.</p> Signup and view all the answers

<p>Maxwell's first equation states that the total electric flux through a closed surface is equal to the total charge inside that surface divided by $rac{1}{ ext{ε}_0}$, relating electric field divergence to charge density.</p> Signup and view all the answers

<p>Maxwell's second equation asserts that the divergence of the magnetic field is zero, indicating that magnetic field lines form closed loops without beginning or end.</p> Signup and view all the answers

<p>The displacement vector, denoted $D$, is defined as $D = ext{ε}E$, which relates electric displacement to the electric field in a material.</p> Signup and view all the answers

<p>In free space, when there are no charges ($P=J=0$), Gauss's law simplifies to $ abla.E = 0$, indicating that the electric field has no divergence.</p> Signup and view all the answers

<p>Faraday's law states that the induced e.m.f. in a circuit is proportional to the negative rate of change of the magnetic flux linked with the circuit, expressed as $e = -rac{d ext{ϕ}_B}{dt}$.</p> Signup and view all the answers

<p>Electric flux, $ ext{ϕ}_E$, is defined as the surface integral of the electric field across a closed surface, capturing the total electric field lines passing through that surface.</p> Signup and view all the answers

<p>If the medium is homogeneous and isotropic, Maxwell's first equation can be expressed as $ abla.( ext{ε}E) = P$, indicating a direct relationship between electric displacement and polarization.</p> Signup and view all the answers

<p>This condition stems from the nature of magnetic field lines forming closed loops, leading to no net magnetic flux through a surface, represented by the equation $ abla .B = 0$.</p> Signup and view all the answers

<p>Stokes's curl theorem relates the line integral of a vector field around a closed curve to the surface integral of the curl of that field over a surface bounded by the curve.</p> Signup and view all the answers

<p>Time-varying magnetic fields cause the curl of the electric field to be non-zero, specifically given by $ abla imes ext{E} = -rac{ ext{dB}}{ ext{dt}}$.</p> Signup and view all the answers

<p>This equation signifies that the curl of the magnetic field is proportional to the current density, linking electric current with the creation of magnetic fields.</p> Signup and view all the answers

<p>A steady current means that the current density does not change with time, allowing the use of Ampere's law without inconsistencies.</p> Signup and view all the answers

<p>Displacement current density is a term added to account for changing electric fields in Maxwell's equations, allowing for the continuity of current in situations where conventional current is not present.</p> Signup and view all the answers

<p>The continuity equation states that the divergence of the current density plus the rate of change of charge density must equal zero, $ abla ext{. J} + rac{ ext{dρ}}{ ext{dt}} = 0$.</p> Signup and view all the answers

<p>Comparing these equations indicates that the divergence of the free current density must be zero, implying that there are no sources or sinks of free current in a steady state.</p> Signup and view all the answers

<p>Maxwell's modification includes both current types to accurately describe the effects of both conduction currents and time-varying electric fields on the magnetic field.</p> Signup and view all the answers

<p>It implies that the displacement current density $ extbf{J_d}$ is related to the rate of change of charge density $ ho$ over time.</p> Signup and view all the answers

<p>Maxwell's modification introduces the displacement current, allowing for the continuity of the electric field in changing conditions and unifying electricity and magnetism.</p> Signup and view all the answers

<p>Stokes's curl theorem relates the line integral of the magnetic field around a closed loop to the surface integral of its curl over the enclosed surface.</p> Signup and view all the answers

<p>The Poynting vector represents the rate of energy flow per unit area in an electromagnetic wave.</p> Signup and view all the answers

<p>It indicates that the curl of the magnetic field $ extbf{B}$ is proportional to the sum of the free current density $ extbf{J_f}$ and the rate of change of the electric displacement field $ extbf{D}$.</p> Signup and view all the answers

<p>Energy density in electromagnetic fields refers to the amount of energy stored per unit volume due to electric and magnetic fields.</p> Signup and view all the answers

<p>For static fields, the condition implies that the curl of the magnetic field $ abla imes extbf{B}$ only depends on the free current density $ extbf{J_f}$.</p> Signup and view all the answers

<p>$ extbf{J_f}$ represents the free current density, while $ extbf{J_d}$ represents the displacement current density, which accounts for changing electric fields.</p> Signup and view all the answers