Maxwell's Equations \[\vec{\nabla}\cdot \vec{E} = \frac{\rho }{\epsilon_{0}} ........(1)\] \[\vec{\nabla}\cdot \vec{B} = 0........(2)\] \[\vec{\nabla}\times \vec{E} = -\frac{\partial \vec{B}}{\partial t}..........(3)\] \[\vec{\nabla}\times \vec{B} = \mu_{0}\vec{J}+\mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t}..........(4)\] This is a traditional way of writing Maxwell's equation, however a bit misleading. Because they reinforce the notion that electric fields can be produced either by charges(ρ) or by changing magnetic fields(∂B/∂t), and magnetic fields can be produced either by currents (J) or by changing electric fields (∂E/∂t). Actually, this is misleading, because ∂B/∂t and ∂E/∂t are themselves due to charges and currents. It will better if we write, \[\vec{\nabla}\cdot \vec{E} = \frac{\rho }{\epsilon_{0}} ........(1)\] \[\vec{\nabla}\cdot \vec{B} = 0........(2)\] \[\vec{\nabla}\times \vec{E} +\frac{\pa...
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Regions in the Complex plane: Neighborhood: A neighborhood consists of all points z lying inside but not on a circle centered at zo and with a specified positive radius ε. When the value of ε is understood or is immaterial in the discussion, the set (1) is often referred to as just a neighborhood. \[\left | z-z_{0} \right |< \epsilon .........(1)\] Deleted Neighborhood: A deleted neighborhood or punctured disk consisting of all points z in an ε neighborhood of z0 except for the point z0 itself. \[0 < \left | z-z_{0} \right |< \epsilon ......(2)\] Interior Point: A point z0 is said to be an interior point of a set S whenever there is some neighborhood of z0 that contains only point...
Impedance - The ratio of a sinusoidal voltage to a sinusoidal current is called "impedance". For a resistor: when current, \[i = e^{j\omega t}\] \[V = iR\] \[V/i = R\] So, the impedance of a resistor is = R For a capacitor: when voltage, \[V = e^{j\omega t}\] \[I = C\frac{dV}{dt}\] \[\Rightarrow I = C j\omega e^{j\omega t}\] \[\Rightarrow I = C j\omega V\] \[\Rightarrow \frac{I}{V} = C j\omega \] \[\Rightarrow \frac{V}{I} = \frac{1}{C j\omega} \] So, the impedance of a capacitor is = \[\frac{1}{C j\omega} \]