Electromagnetic Induction

Moving charges have a magnetic field which influence other moving charges. So current is always combined with a magnetic field. Michael Faraday wanted to know if there is a reverse correlation and observed surprisingly that the existence of a magnetic field will not induce a voltage. But the changing of a magnetic field induced a voltage and if the circuit was closed, a current flew.

This innovative discovery in 1831 from Michael Fraday is called the electro­magnetic induction. It took him nine years to find out. Based on this we do have huge power plants which produce gigawatt hours of electrical energy.

If we move a wire with length $l$ through a magnetic field, electrons experience a Lorentz force. Subsequently charges are seperated and a voltage is induced. If we move the wire with constant velocity $v$, the electric and Lorentz force are equal.
$\begin{align*} F_{el} &= -F_{L}\\ e \cdot V_{ind}/l &= -e \cdot v \cdot B\\ V_{ind} &= -l \cdot v \cdot B\\ \end{align*}$

The magnetic field lines are compressed in the moving direction. With the right hand rule we can determine the direction of current. Lenz's Law: The direction of induced voltage is such, that it will try to oppose the change in flux that is producing it.

If we move a wire loop with the length $l$ through a magnetic field with constant velocity $v$ we can write:

$$ V_{ind}= -l \cdot v \cdot B = -B \cdot l \frac{\Delta x}{\Delta t}$$ $$= -B \cdot\frac{\Delta A}{\Delta t}$$ $$ V_{ind}= -l \cdot v \cdot B = -B \cdot l \frac{\Delta x}{\Delta t} = -B \cdot\frac{\Delta A}{\Delta t}$$

Taking a coil the induced voltage of $N$ loops will be added:

$$V_{ind}=-N \cdot B \cdot \frac{\Delta A}{\Delta t}$$

When switching an electric on or off circuit with a coil and resistor, the magnetic field in the coil changes. So a voltage is induced by itself. When switching on, the voltage direction is oppo­site to the applied voltage so current flow will be blocked. When switch­ing off, the voltage direction is similar, in order to keep the current flowing.

We can compare the coil with an inert millstone. Once running, even if you stop the water, the millstone will still keep turning.

Switching on an electric circuit with a coil and resistor, will result in an exponential function for voltage and current.

$$V_L(t)=V_0 e^{-\frac{t}{\tau}}$$ $$I_L(t)=\frac{V_0}{R} (1-e^{-\frac{t}{\tau}})$$

The time constant $\tau=\frac{L}{R}$ is the elapsed time until the voltage/current has changed to 63 % of its initial value.

Switching off an electric circuit with a coil and resistor, will also result in an exponential function for voltage and current.

$$V_L(t)=-V_0e^{-\frac{t}{\tau}}$$ $$I_L(t)=\frac{V_0}{R}e^{-\frac{t}{\tau}}$$

Self induction is for instance used to switch on a neon tube, since you need a high voltage to ionize the gas inside the tube.

The alternating magnetic field gets trapped in the base of the pane and induces a voltage. The resulting current heats the pan.
Can you explain why the ceramic surface stays relatively cool?

ideas of
G. Vogt, Elektronikschule Tettnang
J. Grehn, J. Krause, "Metzler-Physik", Schroedel Verlag, 2002
K. Johnson et al., "Advanced Physics for You", Oxfor University Press, 2015