Magnetostatics

  1. Biot-Savart’s Law
    1. H-field for finite length of I carrying wire
  2. Ampere’s Circuit Law-Maxwell’s Equation
  3. Magnetic Flux Density - Maxwell’s Equation
  4. Magnetic Scalar and Vector Potentials
  5. Forces due to magnetic fields
  6. Magnetic boundary conditions
  7. Permeability (\(\mu_r\))
  • Magnetostatic field is produced by a constant current flow.
\[\begin{array} {|c|c|c| } \hline \textbf{Term} & \textbf{Electric} & \textbf{Magnetic} \\ \hline \text{basic laws} & \vec{F} = \frac{Q_1Q_2}{4\pi\epsilon^2}\hat{a}_r & d\vec{B} = \frac{\mu I d\vec{l} \times \hat{a}_r}{4\pi R^2} \\ & \int \vec{D} \cdot d\vec{S} = Q_{enc} & \int \vec{H} \cdot d\vec{l} = I_{enc} \\ \hline \text{Force law} & \vec{F} = Q\vec{E} & \vec{F} = Q(\vec{v} \times \vec{B}) \\ \hline \text{Field intensity} & E = \frac{V}{l}\text{(V/m)} & H = \frac{I}{l}\text{(A/m)}\\ \hline \text{Flux density} & D = \frac{\Psi}{S}(\text{C/m}^2) & B = \frac{\Psi}{S} (\text{Wb/m}^2)\\ & \vec{D} = \epsilon\vec{E} & \vec{B} = \mu\vec{H} \\ \hline \text{Potentials} & \vec{E} = -\nabla V & \vec{H} = \nabla V_m \\ & V = \int\frac{\rho_l dl}{4\pi\epsilon} & \vec{A} = \int\frac{\mu I d\vec{l}}{4\pi R} \\ \hline \text{Flux} & \Psi = \int\vec{D}\cdot d\vec{S} & \Psi = \int\vec{B}\cdot d\vec{S}\\ & \Psi = Q = CV & \Psi = LI \\ & I = C\frac{dV}{dt} & V = L\frac{dI}{dt} \\ \hline \text{Energy density} & W_E = \frac{1}{2}\vec{D}\cdot\vec{E} & w_m = \frac{1}{2}\vec{B}\cdot\vec{H}\\ \hline \text{Poisson's eqn} & \nabla^2 V = -\frac{\rho_v}{\epsilon} & \nabla^2 A = -\mu \vec{J}\\ \hline \end{array}\]

Biot-Savart’s Law

  • \(dH \propto \frac{Idl \sin \alpha}{R^2}\) where
    1. \(dH\) = magnetic field intensity at a point P
    2. \(Idl\) = differential current element
    3. \(\alpha\) = angle between current element and line joining it to point P
    4. \(R\) distance between point P and current element

    i.e. \(\fbox{$ d\vec{H} = \frac{Id\vec{l} \times \hat{a}_R}{4\pi R^2} = \frac{Id\vec{l} \times \vec{R}}{4\pi R^3} $}\)

  • direction of magnetic field determined using right-hand screw rule where thumb represents direction of current, rest of the fingers show direction of field.
  • other source elements for surface current density and volume current density are: \(Id\vec{l} \equiv \vec{K}dS \equiv \vec{J}dv\)

H-field for finite length of I carrying wire

hwire

  • \[\vec{H} = \frac{I}{4\pi \rho}(\sin\alpha_1 + \sin\alpha_2) \hat{a}_\phi\]
  • for semi-infinite wire, \(\alpha_1 = 90^\circ, \alpha_2 = 0^\circ, \vec{H} = \frac{I}{4\pi\rho} \hat{a}_\phi\)
  • for infinite wire, \(\alpha_1 = 90^\circ, \alpha_2 = 90^\circ, \vec{H} = \frac{I}{2\pi\rho} \hat{a}_\phi\)

Ampere’s Circuit Law-Maxwell’s Equation

  • If \(I_{enc}\) is current enclosed by a closed path, then \(\fbox{$ \oint \vec{H} \cdot d\vec{l} = I_{enc} $}\)
  • We have \(\oint \vec{H} \cdot d\vec{l} =\int_s (\nabla \times \vec{H}) \cdot d\vec{S} = I_{enc} = \int_s \vec{J} \cdot d\vec{S}\)
    i.e \(\fbox{$ \nabla \times \vec{H} = \vec{J} $}\)
  • \(\nabla \times \vec{H} = \vec{J} \neq 0\) then field is not conservative
  • Infinite line current \(I\) along z-axis \(\vec{H} = \frac{I}{2\pi\rho}\hat{a}_\phi\)
    Infinite sheet current with uniform current density \(\vec{K} = K_y \hat{a}_y\) \(\vec{H} = \frac{1}{2}\vec{K} \times \hat{a}_n\)

Magnetic Flux Density - Maxwell’s Equation

  • magnetic flux density \(\vec{B}\) similar to \(\vec{D}\)
  • \(\fbox{$ \vec{B} = \mu_o \vec{H} $}\) (Webers/square meter) or Teslas
  • magnetic flux \(\psi = \int_s \vec{B} \cdot d\vec{S}\) (Webers)
  • isolated magnetic poles don’t exist hence \(\fbox{$ \oint_s \vec{B}\cdot d\vec{S} = 0 $}\) i.e. \(\fbox{$ \nabla \cdot \vec{B} = 0 $}\)
  1. magnetostatic field is not conserved, but magnetic flux is conserved
  2. magnetic field lines are always continuous
  3. \(\nabla \cdot \vec{b} = 0\) then field is said to be solenoidal

Magnetic Scalar and Vector Potentials

  • Magnetic Scalar Potential \(V_m\) (Amperes) only defined where \(\vec{J} = 0\)
    \(\fbox{$ \vec{H} = -\nabla V_m $}\)
  • vector magnetic potential \(\vec{A}\) (Wb/m) \(\vec{B} = \nabla \times \vec{A}\)
  • scalar potential only used in source-free region

Forces due to magnetic fields

  • charged particle: \(\vec{F} = Q(\vec{E} + \vec{v} \times \vec{B})\)
  • current element: \(d\vec{F} = Id\vec{l} \times \vec{B}\)

Force per unit length between two current carrying conductor is given by \(\fbox{$ \frac{F}{L} = \frac{\mu_o I_1 I_2}{2\pi R} $}\)

Magnetic boundary conditions

  • equations used to determine boundary conditions for magnetic field are: \(\oint \vec{B} \cdot d\vec{S} = 0\) and \(\oint \vec{H} \cdot d\vec{l} = I\)
  • Given \(K\) is surface current at the boundary, \(\theta_1, \theta_2\) are angle to normal of \(\vec{B}_1, \vec{B}_2\) respectively
  • \(\fbox{$ \vec{B}_{1n} = \vec{B}_{2n}$}\) or \(\fbox{$ \mu_1 \vec{H}_{1n} = \mu_2 \vec{H}_{2n}$}\)
  • \(\fbox{$ \vec{H}_{1t} - \vec{H}_{2t} = K $}\) or \(\fbox{$ \frac{\vec{B}_{1t}}{\mu_1} - \frac{\vec{B}_{2t}}{\mu_2}$} = K\)
  • general case \((\vec{H}_1 - \vec{H}_2) \times \hat{a}_{n12} = \vec{K}\)
  • for current free boundary(\(K = 0\)), \(\fbox{$ \frac{\tan\theta_1}{\tan\theta_2} = \frac{\mu_1}{\mu_2} $}\)

Permeability (\(\mu_r\))

Here, external field H, magnetization M, magnetic susceptibility \(\chi\)

  • \(B = \mu_o (H + M)\) and \(B = \mu_r \mu_o H\) i.e. \(M = (\mu_r - 1)H\)
  • where \(\fbox{$ \chi = \mu_r - 1 $}\)