Pointer model

The pointer model is a concept of physics and in particular of physics didactics . It represents periodic processes as the rotation of a pointer and is mainly used in oscillation theory , alternating current theory , wave optics and quantum mechanics .

The pointer usually rotates in the complex plane as a function of time . A fixed, time-independent pointer is used in complex alternating current calculations to explain the phase difference between current and voltage in a circuit with an ohmic resistor, coil and capacitor. Some authors refer to the fixed pointers as phasors and use the versor notation of complex numbers used in technology (Versor = "Dreher").

Basic idea

The vector diagram (left) using the example of a spring pendulum : In the diagram (right), the instantaneous deflection of the pendulum is plotted over time. As you can see, it corresponds to the projection of the pointer onto the y- axis. ${\displaystyle y(t)}$

A pointer of the length rotates at a constant angular velocity around the origin of the coordinates. Its current angle with respect to the axis is denoted by the symbol . ${\displaystyle {\hat {y}}}$ ${\displaystyle \omega }$ ${\displaystyle x}$ ${\displaystyle \varphi (t)}$

If you illuminate this pointer parallel to the axis with a lamp, it casts a shadow of the length on a vertical wall. The simple trigonometric relationship applies ${\displaystyle x}$ ${\displaystyle y(t)}$

${\displaystyle y(t)={\hat {y}}\cdot \sin(\varphi (t))={\hat {y}}\cdot \sin(\omega t+\varphi _{0})}$

where is the starting angle. ${\displaystyle \varphi _{0}}$

The change in the shadow is a harmonic vibration . The sizes used have the following meanings:

Formula symbol	unit	Meaning in the pointer model	Meaning for the vibration
${\displaystyle {\hat {y}}}$	any	Length of the pointer	Amplitude of the oscillation
${\displaystyle y(t)}$	any	"Shadow" of the pointer	Momentary deflection
${\displaystyle \varphi (t)}$	${\displaystyle \mathrm {rad} }$	Current angle	Phase angle
${\displaystyle \varphi _{0}}$	${\displaystyle \mathrm {rad} }$	Starting angle	Zero phase angle
${\displaystyle \omega =2\pi f}$	${\displaystyle \mathrm {\frac {rad}{s}} }$	Angular velocity	Angular frequency
${\displaystyle f={\frac {1}{T}}}$	${\displaystyle \mathrm {\frac {1}{s}} }$	rotational speed	frequency
${\displaystyle T}$	${\displaystyle \mathrm {s} }$	Period of circulation	Period duration

Complex number level

Illustration in the complex number plane

The pointer model is often represented in the complex plane of numbers. The pointer is then a complex quantity ${\displaystyle {\underline {z}}}$

${\displaystyle {\underline {z}}=x+\mathrm {i} y}$

with the real part and the imaginary part . With Euler's formula , the deflection-time law of the oscillation can be written as follows: ${\displaystyle \mathrm {Re} \,{\underline {z}}=x}$ ${\displaystyle \mathrm {Im} \,{\underline {z}}=y}$

${\displaystyle {\underline {z}}(t)={\hat {z}}\cdot \mathrm {e} ^{\mathrm {i} (\omega t+\varphi _{0})}={\hat {\underline {z}}}\cdot \mathrm {e} ^{\mathrm {i} \omega t}}$

The complex quantity is sometimes referred to as the phasor or "complex amplitude". If one takes only the imaginary part, one arrives at an equation like from the previous section. But you can just as easily work with the real part. The cosine oscillation then takes the place of the sine oscillation. Since the sine and cosine functions only differ in the constant phase shift angle of , both mathematical formulations are equivalent; within a problem, however, one must either choose one or the other representation. ${\displaystyle {\hat {\underline {z}}}={\hat {z}}\cdot \mathrm {e} ^{\mathrm {i} \varphi _{0}}}$ ${\displaystyle {\underline {z}}(t)}$ ${\displaystyle {\tfrac {\pi }{2}}}$

Applications

Electrical engineering: alternating current theory

Alternating current and voltage in the pointer model, here the example of a series circuit of resistance R and the inductance L . The pointers of u and i rotate around the origin of the coordinates. The real, reactive and apparent resistances (black arrows, they form the " resistance triangle ") do not change.

→ Main article : Complex AC bill

In alternating current theory, one considers the sinusoidal alternating voltage and the sinusoidal alternating current strength . Both can be represented as pointers, which rotate together with the angular velocity around the origin of the coordinates and have the constant phase shift angle . ${\displaystyle {\underline {u}}(t)={\hat {u}}\cdot e^{\mathrm {j} (\omega t+\varphi _{u})}}$ ${\displaystyle {\underline {i}}(t)={\hat {\imath }}\cdot e^{\mathrm {j} (\omega t+\varphi _{i})}}$ ${\displaystyle \omega }$ ${\displaystyle \Delta \varphi =\varphi _{u}-\varphi _{i}}$

If one analogously to the relationship that holds for direct currents, the equation ${\displaystyle R={\frac {U}{I}}}$

${\displaystyle {\underline {Z}}={\frac {{\underline {u}}(t)}{{\underline {i}}(t)}}}$

for alternating currents and voltages, one obtains the impedance , the amount of which is also called "impedance". Note that the impedance is not time-dependent, because the factor is canceled out. In the general case, however, it is complex-valued: ${\displaystyle e^{\mathrm {j} \omega t}}$

${\displaystyle {\underline {Z}}=R+\mathrm {j} X}$

The real part is the ohmic resistance or effective resistance . The imaginary part is called reactance . It is made up of ${\displaystyle R}$ ${\displaystyle X}$

• the inductive reactance and ${\displaystyle X_{L}=\omega L}$
• the capacitive reactance ${\displaystyle X_{C}=-{\frac {1}{\omega C}}}$

The advantage of displaying sinusoidal alternating current quantities as complex vectors in the alternating current diagram is that the essential laws of electricity theory (use of impedance like a resistor, Kirchhoff's rules ) can also be used in alternating current theory without the need for complicated trigonometric calculations.

Note: The pointer length represents the absolute value of voltage and current. In practice, instead of the amplitude Û and Î ("amplitude pointer "), the rms value U and I ("rms value pointer ") is often used.

Wave optics

In the one-dimensional case, a sine wave is described by the following equation:

${\displaystyle s(x,t)={\hat {s}}\cdot \sin(\omega t-kx)}$

Where is the circular wave number . The zero phase angle should be zero for the sake of simplicity. ${\displaystyle k}$ ${\displaystyle k={\frac {2\pi }{\lambda }}}$

Here, too, one can imagine the momentary deflection by a rotating pointer, whereby this time the angle depends not only on the time, but also on the location. If you look at the shaft at a location that is one wavelength away from the origin, the pointer has covered one revolution less at this location than a pointer in the origin of the coordinates. So one has on the angle of the respective pull-fold of the distance. ${\displaystyle \omega t}$ ${\displaystyle k}$

interference

→ Main article : Interference (physics)

Interference in the pointer model: This diagram shows a snapshot of two sine waves (red and blue) of the same amplitude and frequency that are interfering. At a certain point the pointers of both waves were drawn as an example.

If two waves overlap at one point, the pointers of both waves must be added vectorially , as shown in the adjacent figure as an example for a point. The instantaneous deflection of the resulting oscillation is then obtained again by projecting the resulting (violet) pointer onto the vertical axis drawn at the desired point. The length of this pointer also indicates the amplitude of the resulting wave (purple line). In addition to the amplitudes of the waves involved, their phase difference is also decisive for the result of the interference . This is particularly easy with waves of the same frequency, since the phase difference is constant here. ${\displaystyle \Delta \varphi }$

The following applies:

• ${\displaystyle \Delta \varphi =0,\,2\pi ,\,4\pi ,\,\dots }$: Constructive interference. The amplitudes of the two waves add up.
• ${\displaystyle \Delta \varphi =\pi ,\,3\pi ,\,5\pi ,\,\dots }$: Destructive interference. The amplitudes of the two waves must be subtracted from each other. If they are the same, they cancel one another out.

If several waves overlap at one point, the pointers of all waves must be added vectorially.

Standing waves

→ Main article : Standing wave

Standing waves. Two opposing sine waves interfere. The red wave runs to the right. The blue wave runs to the left. The purple pointers and lines show a snapshot of the resulting wave. The dashed purple lines represent the maximum deflections. Explanation s. Text.

If two opposing waves of the same frequency are superimposed, a standing wave is created. In the illustration opposite, the red wave runs to the right, the blue wave to the left. If you pick out a certain point, the pointers of the two waves have a certain phase difference. This difference does not depend on the time, as both hands rotate at the same speed in the same direction. Still, it depends on the place. At locations where the phase difference is or - that is, where the two waves are in phase - the instantaneous deflection is always maximum compared to other locations. This is called an "antinode". There is no deflection at all at the points where the phase difference is. This is called a "vibration node". Since neither the antinodes nor the nodes move, it appears as if the wave is not spreading at all, hence the name "standing wave". The maximum deflection of the standing wave at an antinode is given by the sum of the pointer lengths, i.e. the sum of the amplitudes. ${\displaystyle 0}$ ${\displaystyle 2\pi }$ ${\displaystyle \pm \pi }$

diffraction

→ Main article : Diffraction (physics)

In the case of multi-dimensional problems (e.g. single slit , double slit , optical grating , ...) it must be taken into account that waves that meet at one point can have traveled different paths. The path differences are then calculated . A path difference is equivalent to a phase difference of . The diffraction pattern is thus obtained by vector addition of the pointers of the interfering waves, taking into account the phase difference resulting from the path difference. ${\displaystyle \delta }$ ${\displaystyle \Delta \varphi =k\delta }$

Limits

While phase differences and their effects on the interference can be explained very well with the pointer model, the calculation of the amplitudes fails because neither the attenuation nor the distribution of a wave in space can be taken into account by the pointer model. Alternative concepts such as: B. the elementary waves according to Huygens and Fresnel .

Quantum mechanics

The wave function of quantum mechanics can also be represented in the pointer model. Feynman calls the (complex) length of the pointer the "probability amplitude", since its square is a measure of the probability density according to the rules of quantum mechanics (e.g. for finding a particle). This also leads to the effect of interference, as described in the section on wave optics above. When a quantum object goes through an experimental set-up, the probability amplitudes for all possible paths must be added vectorially. With this Feynman finds a clear interpretation for the method of path integrals .

Web links

• Extensive collection of animations (mostly Java applets), some of which use the pointer model

Individual evidence, receipts and comments

1. ^ W. Philipp: Pointer model in physics lessons at the course level. ( Memento of the original from September 4, 2013 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. (pdf) 
2. ↑ In this way, the vibrations are introduced in many school books for the upper school level, e.g. B. in Dorn, Bader: Physics Gymnasium (G8) 11/12. Schroedel, 2010, ISBN 978-3-507-10748-9 ; Meyer, Schmitt: textbook physics, high school upper level. Duden, 2011, ISBN 978-3-8355-3311-0 ; Boysen among others: Upper level physics complete volume. Cornelsen, 1999, ISBN 3-464-03440-2 .
3. ↑ In the theory of alternating currents, the imaginary unit is written as to avoid confusion with the current strength. ${\displaystyle \mathrm {j} }$
4. ↑ In the section "Wave Optics" - as usual in school physics - a representation with real pointers is used. When working with complex numbers, the sine function is replaced by the complex exponential function, as described in the section “Complex number plane”. The vector addition used here corresponds to the addition of complex numbers.
5. ^ Feynman, Leighton, Sands: Feynman lectures on physics. Volume III: Quantum Physics. , 4th edition. Oldenbourg, 1999, ISBN 3-486-25134-1 , pp. 29/39.
6. ^ R. Feynman: QED The strange theory of light and matter. 8th edition. Piper, 2002, ISBN 3-492-21562-9 .