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## Induction when the magnetic field changes

### Induction in the ladder swing

The arrangement from the already known ladder swing experiment is now modified so that the voltage source is replaced by a voltage measuring device. The hanging metal rod is therefore initially not traversed by a current. If you approach the horseshoe magnet quickly, you can see a short deflection on the voltmeter. After the horseshoe magnet has come to rest, no voltage is displayed as before. If you remove the horseshoe magnet again with a quick movement, a voltage surge is displayed again, this time with the opposite sign.

Explanation: The stationary magnet has no effect on the ladder swing. However, if the magnet is moved towards or away from the ladder swing, a voltage is induced during the movement, which is registered by the tension meter as a brief deflection.

### Induction in a coil (by a bar magnet)

If the ends of a coil are connected to a voltage measuring device, no deflection of the device is initially observed. If a bar magnet is approached to the coil, a voltage is displayed during the movement. If the bar magnet is kept at rest, the deflection goes back to zero. When the bar magnet is removed, a voltage is again induced which is opposite to the previous deflection.

If the bar magnet is always moved into the same proximity to the coil at different speeds, it can also be seen that the faster the magnet is moved, the greater the induced voltage.

In summary, one observes:

- When the magnet approaches, a voltage is induced.
- When removed, an oppositely polarized voltage is induced.
- When the magnet is at rest, no voltage is induced, even if it is in the immediate vicinity of the coil.
- The faster the magnet is moved, the greater the induced voltage.

Explanation: A changing magnetic field induces a voltage in a conductor loop. A constant magnetic field has no effect on the conductor loop.

### Induction in a coil (by another coil)

In order to be able to quantitatively investigate the relationship between the change in the magnetic field and the induced voltage, the magnetic field of the bar magnet can be replaced by the magnetic field of another coil. The conductor loop is replaced by a small coil (induction coil), which is inserted into a larger coil (field coil) with exactly the same orientation.

Since the magnetic field inside the field coil is known as a function of its dimensions and the current flowing through its windings, a quantitative relationship can now be established between the coil magnetic field or the change in this magnetic field and the induced voltage in the induction coil.

A linearly increasing and linearly decreasing current can alternately be generated in the field coil by means of a triangular generator. Because$\mathrm{B.}={\mu}_{0}\cdot \mathrm{I.}\cdot n/l$ the magnetic field of the coil is then also correspondingly linearly increasing or decreasing. A linear increase or decrease means that the change $d\mathrm{I.}/dt$ respectively. $d\mathrm{B.}/dt$ is constant.

In the ascent phase ($d\mathrm{B.}/dt=\text{const.}>0$) the voltmeter shows a constant, positive induction voltage and, in the phase of falling, a constant negative induction voltage. If the triangle generator generates a symmetrical triangular voltage, the magnitude of the induction voltage is the same in both phases. The greater the increase in the triangular voltage, the greater the magnitude of the induced voltage.

So one observes

- $${U}_{ind}\sim \frac{\text{d}\mathrm{B.}}{\text{d}t}\text{.}$$

This result is also obtained with non-constant changes in the magnetic field$d\mathrm{B.}/dt$. For example, the triangular current is replaced by a sinusoidal current $\mathrm{I.}(t)={\mathrm{I.}}_{0}\cdot sin(\omega t)$ replaced, the magnetic field strength of the field coil also changes sinusoidally $\mathrm{B.}(t)={\mathrm{B.}}_{0}\cdot sin(\omega t)$ . The voltage induced in the induction coil is then proportional to the time derivative of this sine function: ${U}_{ind}(t)\sim {\mathrm{B.}}_{0}\cdot cos(\omega t)$ .

### Change in the number of turns in the induction coil

With an unchanged change in the magnetic field $d\mathrm{B.}/dt$ the number of turns of the induction coil is changed. This can be achieved simply by tapping the induction voltage over a fraction of the turns (1 / 2.1 / 3, 1/4). In this way, other parameters (e.g. the cross section of the induction coil) remain unchanged.

If the induction voltage is tapped from only one half of the induction coil, the amount of the induction voltage is halved. The same can be observed for any fraction of the turns used. We can conclude that the induced voltage is proportional to the number $n$ of the turns of the induction coil.

- $${U}_{ind}\sim n\cdot \frac{\text{d}\mathrm{B.}}{\text{d}t}$$

### Change in cross-sectional area of the induction coil

Next, induction coils of different cross-sectional areas with the same number of turns are used $n$ and the same change in magnetic field $d\mathrm{B.}/dt$ considered.

If the two coils are parallel to each other and the cross-sectional area of the induction coil is halved, the induced voltage is halved. If the cross-sectional area is doubled, the voltage is doubled. The induction voltage is proportional to the cross-sectional area $\mathrm{A.}$ the induction coil:

- law
- Induced voltage with constant cross-sectional area and changing magnetic field.

- $${U}_{ind}=n\cdot \mathrm{A.}\cdot \frac{\text{d}\mathrm{B.}}{\text{d}t}$$

### Angle dependence

If the induction coil is placed in the field coil at an angle, the induction voltage decreases with otherwise unchanged conditions. This leads us to believe that the orientation of the face of the induction coil to the direction of the magnetic field plays a role.

We denote the angle between the two coil axes with $\alpha $ . Is the induction coil perpendicular to the field coil ( $\alpha $ = 90 °), the induction voltage approaches zero. Shows the induction coil parallel to the field coil ($\alpha $ = 0 °) then the induction voltage (if the other parameters remain the same) assumes its maximum value. For other positions one observes a steady change in the voltage between zero and maximum value as a function of the angle $\alpha $ :

- $${U}_{ind}\sim cos(\alpha )\text{.}$$

The reason is that it is not the absolute cross-sectional area that is decisive for the magnitude of the induction voltage, but the vertical projection of the area in the direction of the field. Or, to put it more clearly: The area through which the field penetrates vertically, which is

- $${\mathrm{A.}}_{\perp}=\mathrm{A.}\cdot cos(\alpha )\text{.}$$

With this we get:

- $${U}_{ind}=n\cdot {\mathrm{A.}}_{\perp}\cdot \frac{\text{d}\mathrm{B.}}{\text{d}t}=n\cdot \mathrm{A.}\cdot cos(\alpha )\cdot \frac{\text{d}\mathrm{B.}}{\text{d}t}\text{.}$$

### Area vector

To make this clearer, the area of the is sometimes used as a vector $\overrightarrow{\mathrm{A.}}$ written, i.e. as a quantity with an amount and a direction. The amount of size$\overrightarrow{\mathrm{A.}}$ is simply the amount of the area. The direction of $\overrightarrow{\mathrm{A.}}$ is defined as the normal unit vector on the surface, i.e. pointing perpendicular to the surface. As an angle $\alpha $ we can then consider the angle between the area vector and the field strength vector.

The relationship can thus be formulated vectorially:

- law
- Induced voltage with constant cross-sectional area and changing magnetic field.

- $${U}_{ind}=n\cdot \overrightarrow{\mathrm{A.}}\cdot \frac{\text{d}\overrightarrow{\mathrm{B.}}}{\text{d}t}\text{.}$$

So far we have considered the induction voltage with a constant cross-sectional area and a changing magnetic field. In the following we clarify the dependence of the induction voltage on a change in area at a constant magnetic field.