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Electromagnetism

Characteristics of Coil, Resistor and Capacitor

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At university we often think of series RLC circuits. L is a coil, R is a resistance, and C is a capacitor. The relationship between the voltage applied to each electronic component and the current is given as follows.

Voltage of Coil \(V_L\)

$$V_{L}(t)=L\frac{dI(t)}{dt}$$

Voltage of Resistance \(V_R\)

$$V_{R}(t)=RI(t)$$

Voltage of Capacitor \(V_C\)

$$V_{C}(t)=\frac{1}{C} \int I(t) dt=\frac{Q(t)}{C}$$

\(L\):Self-inductance of the coil　\(R\):Resistance   \(C\):Capacitance   \(Q(t)\):Charge stored in the capacitor

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Differential Form and Integral Form of Maxwell Equation

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Differential Form of Maxwell Equation

Gauss’s law

$$div {\bf D}({\bf r},t)=ρ({\bf r},t)$$

Law of the magnetic flux conservation

$$div {\bf B}({\bf r},t)=0$$

Maxwell-Ampere law

$$rot {\bf H}({\bf r},t)={\bf i}({\bf r},t)+\frac{∂{\bf D}({\bf r},t)}{∂t}$$

Faraday’s law of induction

$$rot {\bf E}({\bf r},t)=-\frac{∂{\bf B}({\bf r},t)}{∂t}$$

Relation between electric field and electric flux density, magnetic field and magnetic flux density

$${\bf D}({\bf r},t)=ε{\bf E}({\bf r},t)$$
$${\bf B}({\bf r},t)=μ{\bf H}({\bf r},t)$$

\({\bf E}({\bf r},t)\):Electric field   \({\bf D}({\bf r},t)\):Electric flux density   \({\bf H}({\bf r},t)\):Magnetic field   \({\bf B}({\bf r},t)\):Magnetic flux density

\(ρ({\bf r},t)\):Charge density   \({\bf i}({\bf r},t)\):Current density   \(ε\):Dielectric constant (Dielectric)   \(μ\):Magnetic permeability (Permeability)

The first four equations are collectively called the Maxwell equation. The form is called the differential form of the Maxwell equation. The latter two supplements wrote the electric field and electric flux density, and the relational expressions of the magnetic field and the magnetic flux density, respectively.

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