AEiE0601 Electric field and magnetic field¶
Field quantities and units¶
| Symbol | Quantity | Meaning | SI unit |
|---|---|---|---|
| \(E\) | Electric field intensity | Force per unit positive test charge | V/m or N/C |
| \(D\) | Electric flux density | Flux per unit area, includes material effect | C/m\(^2\) |
| \(H\) | Magnetic field intensity | Magnetizing force | A/m |
| \(B\) | Magnetic flux density | Magnetic flux per unit area | tesla (T) = Wb/m\(^2\) |
Mandatory distinction:
- \(E\) and \(D\) are not the same quantity.
- \(H\) and \(B\) are not the same quantity.
- In linear isotropic media, \(D=\epsilon E\) and \(B=\mu H\).
Fundamental relations¶
\[
D=\epsilon E, \qquad B=\mu H.
\]
For free space:
\[
\epsilon_0 \approx 8.854\times10^{-12}\ \text{F/m}, \qquad
\mu_0=4\pi\times10^{-7}\ \text{H/m}.
\]
Relative material parameters:
\[
\epsilon=\epsilon_r\epsilon_0, \qquad \mu=\mu_r\mu_0.
\]
Divergence and divergence theorem¶
- Divergence measures net outward flux generation per unit volume.
- Positive divergence suggests a source region; negative divergence suggests a sink region.
- Gauss divergence theorem converts a volume integral of divergence into a closed-surface flux integral:
\[
\iiint_V (\nabla\cdot A)\,dV = \iint_S A\cdot dS.
\]
Electrostatic cue:
\[
\nabla\cdot D = \rho_f
\]
where \(\rho_f\) is free volume-charge density.
Electric potential and potential gradient¶
- Electric potential \(V\) is electric potential energy per unit charge.
- Electric field is the negative gradient of potential:
\[
E=-\nabla V.
\]
- Field points in the direction of greatest potential decrease.
The potential difference is
\[
V_b-V_a=-\int_a^b E\cdot d\ell.
\]
Energy density in electrostatic field¶
\[
w_e=\frac{1}{2}E\cdot D = \frac{1}{2}\epsilon E^2
\]
for a linear isotropic medium.
Free charge, bound charge, and polarization¶
- Free charge is charge not represented by material polarization; it may be mobile or fixed.
- Bound charges arise from material polarization, with volume and surface densities
\[
\rho_b=-\nabla\cdot P, \qquad \sigma_b=P\cdot\hat n.
\]
- Polarization vector \(P\) represents electric dipole moment per unit volume.
- Relative permittivity \(\epsilon_r\) indicates how strongly the material polarizes compared with free space.
Electric dipole¶
- Two equal and opposite charges separated by a small distance form an electric dipole.
- Dipole moment magnitude:
\[
p=qd
\]
with direction from negative charge to positive charge.
Electric boundary conditions¶
At a boundary between two media:
- Tangential component of \(E\) is continuous if there is no surface time-varying magnetic effect in electrostatics.
- Normal component of \(D\) changes by free surface charge density:
\[
D_{2n}-D_{1n}=\rho_s.
\]
Special conductor cues:
- Inside an electrostatic perfect conductor, \(E=0\).
- Electric field at the conductor surface is normal to the surface in electrostatics.
Magnetic quantities and effects¶
- Magnetic force on a moving charge:
\[
F=q(v\times B).
\]
- Force on a current-carrying conductor of length vector \(\ell\):
\[
F=I(\ell\times B).
\]
- Torque on a magnetic dipole in field \(B\):
\[
\tau = m\times B.
\]
where \(m\) is magnetic dipole moment.
Magnetization and magnetic dipole¶
- Magnetization \(M\) is magnetic dipole moment per unit volume.
- A magnetic dipole can be modeled by a tiny current loop.
- In linear media, magnetization contributes to the relation between \(B\) and \(H\).
Magnetic boundary conditions¶
- Normal component of magnetic flux density is continuous:
\[
B_{1n}=B_{2n}.
\]
- Tangential component of magnetic field intensity changes by free surface current density:
\[
\hat n\times(H_2-H_1)=K_f
\]
where \(\hat n\) points from medium 1 to medium 2 and \(K_f\) is free surface current density.
Field examples¶
- If the unit is C/m\(^2\), the quantity is \(D\), not \(E\).
- If the unit is tesla, the quantity is \(B\), not \(H\).
- If the field is derived from potential, use \(E=-\nabla V\).
AEiE0601 revision box¶
- \(D=\epsilon E\), \(B=\mu H\).
- Units: \(E\) in V/m, \(D\) in C/m\(^2\), \(H\) in A/m, \(B\) in T.
- Divergence theorem links closed-surface flux to volume divergence.
- \(E=-\nabla V\).
- Normal \(D\) jump equals free surface charge; normal \(B\) is continuous.