6.1 Faraday’s Law

Magnetic fields can produce an electric current in a closed loop, but only if the magnetic flux linking the surface area of the loop changes with time. The key to induction is change.

$$ \Phi = \int _s B \cdot ds $$

• Induced electromotive force through a process called electromagnetic induction in a closed conducting loop of $N$ turns is given by:

  $$ V_{emf} = -N \frac{d \Phi}{dt} = -N \frac{d}{dt} \int _s B \cdot ds $$

Conditions where EMF can be generated in a closed conducting loop:

1. Time-varying magnetic field linking a stationary loop; the induced EMF is called the transformer EMF, $V_{emf}^{tr}$
2. A moving loop with a time-varying area (relative to the normal component of B) in a static field B
   1. The induced EMF is then called the motional EMF, $V_{emf} ^{m}$
3. A moving loop in a time-varying field B.

Total EMF is given by:

$$ V_{emf} = V_{emf} ^{tr}+V_{emf}^{m} $$

6.2 Stationary Loop in a Time-Varying Magnetic Field

$$ V_{emf}^{tr} = -N \int _s \frac{\delta B}{\delta t} \cdot ds \text{, transformer EMF} $$

Loop in a charging $B$ field

If loop has an internal resistance $R_i$, the circuit can be represented by the equivalent circuit below:

Equivalent circuit.

• The current $I$ flowing through the circuit is given by:

  $$ I = \frac{V_{emf}^{tr}}{R+R_i} $$

• The polarity of $V_{emf}^{tr}$ and the direction of $I$ is governed by Lenz’s law, which states that the current in the loop is always in a direction that opposes the change of magnetic flux $\Phi (t)$ that produced $I$

• $B_{ind}$ serves to opposte the change in $B(t)$ and not necessarily $B(t)$ itself

Faraday’s Law

$$ \nabla \times E = -\frac{ \delta B}{\delta t} $$

6.3 The Ideal Transformer