Electromagnetism - Maxwell's Equations & Vector Fields Flashcards

Question 1

Poynting Vector

Accepted Answer

$\mathbf{S} = \frac{1}{\mu_0} (\mathbf{E} \	imes \mathbf{B})$

Represents the directional energy flux density (power flow per unit area, measured in $\	ext{W/m}^2$) of an electromagnetic field.

Question 2

Gauss's Law for Electricity (Differential Form)

Accepted Answer

$\
abla \cdot \mathbf{E} = \frac{\ho}{\epsilon_0}$

States that the divergence of the electric field $\mathbf{E}$ at any point is proportional to the local volume charge density $\ho$. Electric charges act as sources or sinks for electric field lines.

Question 3

Gauss's Law for Magnetism (Differential Form)

Accepted Answer

$\
abla \cdot \mathbf{B} = 0$

States that the divergence of the magnetic field $\mathbf{B}$ is always zero. This implies there are no magnetic monopoles; magnetic field lines must form continuous closed loops.

Question 4

Faraday's Law of Induction (Differential Form)

Accepted Answer

$\
abla \	imes \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$

States that a time-varying magnetic field induces a circulating, non-conservative electric field. The negative sign represents Lenz's Law, indicating the induced field opposes the change in magnetic flux.

Question 5

Ampère-Maxwell Law (Differential Form)

Accepted Answer

$\
abla \	imes \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

States that magnetic fields are generated both by free electric current density $\mathbf{J}$ and by time-varying electric fields (displacement current).

Question 6

Displacement Current Density

Accepted Answer

$\mathbf{J}_d = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

A term added by Maxwell to Ampère's Law to maintain conservation of electric charge in time-varying fields, resolving inconsistencies in circuit models like charging capacitors.

Question 7

Continuity Equation (Charge Conservation)

Accepted Answer

$\
abla \cdot \mathbf{J} + \frac{\partial \ho}{\partial t} = 0$

Expresses local conservation of charge: any decrease in local charge density $\ho$ over time must equal the net outward current flux density $\mathbf{J}$ exiting that volume.

Question 8

Speed of Light from Fundamental Constants

Accepted Answer

$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$

Derived directly from Maxwell's equations in free space. The speed of electromagnetic waves in a vacuum depends solely on the vacuum permittivity $\epsilon_0$ and permeability $\mu_0$ ($c \approx 3 \	imes 10^8 \	ext{ m/s}$).

Question 9

Electromagnetic Wave Equation for Electric Field

Accepted Answer

$\
abla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}$

Derived in source-free space ($\ho = 0, \mathbf{J} = 0$) by taking the curl of Faraday's Law and applying the vector identity $\
abla \	imes (\
abla \	imes \mathbf{E}) = \
abla(\
abla \cdot \mathbf{E}) - \
abla^2 \mathbf{E}$.

Question 10

Boundary Condition: Normal Component of $\mathbf{D}$

Accepted Answer

$D_{1n} - D_{2n} = \sigma_f$ or $\hat{\mathbf{n}} \cdot (\mathbf{D}_1 - \mathbf{D}_2) = \sigma_f$

The normal component of the electric displacement field $\mathbf{D}$ is discontinuous across a interface by an amount equal to the free surface charge density $\sigma_f$.

Question 11

Boundary Condition: Tangential Component of $\mathbf{E}$

Accepted Answer

$E_{1t} = E_{2t}$ or $\hat{\mathbf{n}} \	imes (\mathbf{E}_1 - \mathbf{E}_2) = 0$

The tangential component of the electric field $\mathbf{E}$ is continuous across any interface between two media.

Question 12

Boundary Condition: Normal Component of $\mathbf{B}$

Accepted Answer

$B_{1n} = B_{2n}$ or $\hat{\mathbf{n}} \cdot (\mathbf{B}_1 - \mathbf{B}_2) = 0$

The normal component of the magnetic field $\mathbf{B}$ is continuous across any boundary because magnetic monopoles do not exist.

Question 13

Boundary Condition: Tangential Component of $\mathbf{H}$

Accepted Answer

$\hat{\mathbf{n}} \	imes (\mathbf{H}_1 - \mathbf{H}_2) = \mathbf{K}_f$

The tangential component of the magnetic field intensity $\mathbf{H}$ is discontinuous across an interface by an amount equal to the free surface current density $\mathbf{K}_f$.

Question 14

Lorentz Force Law

Accepted Answer

$\mathbf{F} = q(\mathbf{E} + \mathbf{v} \	imes \mathbf{B})$

Describes the total electromagnetic force $\mathbf{F}$ exerted on a particle with charge $q$ moving at velocity $\mathbf{v}$ through electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields.

Question 15

Magnetic Vector Potential ($\mathbf{A}$)

Accepted Answer

A vector field $\mathbf{A}$ defined such that $\mathbf{B} = \
abla \	imes \mathbf{A}$.

Since $\
abla \cdot \mathbf{B} = 0$ everywhere, $\mathbf{B}$ can always be expressed as the curl of a vector field using the vector identity $\
abla \cdot (\
abla \	imes \mathbf{A}) = 0$.

Question 16

Electric Field in Terms of Scalar and Vector Potentials

Accepted Answer

$\mathbf{E} = -\
abla V - \frac{\partial \mathbf{A}}{\partial t}$

In dynamic (time-varying) conditions, the electric field depends on both the electric scalar potential $V$ and the time derivative of the magnetic vector potential $\mathbf{A}$.

Question 17

Lorenz Gauge Condition

Accepted Answer

$\
abla \cdot \mathbf{A} + \mu_0 \epsilon_0 \frac{\partial V}{\partial t} = 0$

A gauge condition chosen to decouple the differential equations for $V$ and $\mathbf{A}$, producing symmetric, relativistic wave equations for both potentials.

Question 18

Coulomb Gauge Condition

Accepted Answer

$\
abla \cdot \mathbf{A} = 0$

A gauge condition where the scalar potential satisfies Poisson's equation $\
abla^2 V = -\frac{\ho}{\epsilon_0}$, allowing instant calculation of $V$, widely used in electrostatics and quantum optics.

Question 19

Divergence Operator Interpretation

Accepted Answer

$\
abla \cdot \mathbf{V} = \lim_{\Delta V \	o 0} \frac{1}{\Delta V} \oint_S \mathbf{V} \cdot d\mathbf{a}$

Measures the net outward flux of a vector field per unit volume from an infinitesimal point. A positive divergence signifies a net source; negative divergence signifies a net sink.

Question 20

Curl Operator Interpretation

Accepted Answer

$(\
abla \	imes \mathbf{V}) \cdot \hat{\mathbf{n}} = \lim_{\Delta A \	o 0} \frac{1}{\Delta A} \oint_C \mathbf{V} \cdot d\mathbf{l}$

Measures the net circulation or local rotational density of a vector field per unit area around a specific axis.

Electromagnetism - Maxwell's Equations & Vector Fields

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