a field can only be electric or magnetic of it can satisfy corresponding maxwell’s equation
above equations are for static field. The divergence equations remain same but the curl equations change for time-varying fields.
Faraday’s Law of Induction
time-varying current was obtained in the closed wire loop while magnet was being moved toward it or away from it.
It states that the line integral of electric field around a closed loop equal the rate of change of magnetic flux through the loop.
Right-hand thumb rule if curling of fingers point in direction of traversal in the loop then thumb points in the direction of positive magnetic flux.
For any loop, \(\fbox{$ V_{emf} = \oint_L \vec{E} \cdot d\vec{l} = -\frac{d\phi}{dt} = - \frac{d}{dt} \int_s \vec{B}\cdot d\vec{S} $}\)
where EMF is electromotive force
Transformer and Motional EMFs
The variation in flux with time can be caused in three ways:
1. Stationary Loop in time-varying B-Field(Transformer EMF)
This is referred to transformer emf in power analysis since it is due to transformer action.
hence, Faraday’s Law of induction in differential form:
\(\fbox{$ \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} $}\)
time-varying E-field is not conservative \(\nabla \times \vec{E} \neq 0\)
2. Moving loop in static B-Field (Motional EMF)
emf is induced in conduction loop when moved in a static B-field. It is called motional because it is due to motion. Found in motors, generators and alternators.
We define motional E-field as \(\vec{E}_m = \frac{\vec{F}_m}{Q} = \vec{v} \times \vec{B}\)
i.e. \(\fbox{$ \nabla \times \vec{E} = \nabla \times (\vec{v} \times \vec{B}) $}\)
3. Moving loop in time-varying fields
this is the general case
i.e. \(\fbox{$ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} + \nabla \times (\vec{v} \times \vec{B}) $}\)
Displacement Current
For static EM field, \(\nabla \times \vec{H} = \vec{J}\)
And divergence of curl is zero. i.e. \(\nabla \cdot (\nabla \times \vec{H}) = \nabla \cdot \vec{J} = 0\)
However, continuity equation is given by \(\fbox{$ \nabla \cdot \vec{J} = - \frac{\partial \rho_v}{\partial t} $}\)
Displacement current cannot be zero, so to the static EM eqn compatible we add a term \(\vec{J}_d\) which is determined by using the fact that divergence of curl of any vector is zero.
\(\begin{align*}
\nabla \times \vec{H} &= \vec{J} + \vec{J}_d \\
\nabla \cdot (\nabla \times \vec{H}) &= \nabla \cdot \vec{J} + \nabla \cdot\vec{J}_d = 0 \\
\nabla \cdot \vec{J}_d &= -\nabla \cdot\vec{J} = \frac{\partial \rho_v}{\partial t} = \frac{\partial }{\partial t}(\nabla \cdot \vec{D}) = \nabla \cdot \frac{\partial \vec{D}}{\partial t} \\
\vec{J}_d &= \frac{\partial \vec{D}}{\partial t}
\end{align*}\)
Finally we have \(\fbox{$ \nabla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}$}\)
where \(\vec{J}_d\) is displacement current density
and \(\vec{J} = \sigma\vec{E}\) is conduction current density
Maxwell’s Equation in final form
for a field to be qualified as EM field, it must satisfy all four equations.