Induction
This program can be used to study induction on an open conductor loop. Faraday's law of induction predicts that the induced voltage is equal to the negative derivative of the magnetic flux with respect to time. This law contains two mechanisms that lead to the occurrence of the induced voltage. Firstly: If the magnetic field strength B (old term: magnetic flux density) changes, an electric field is generated which sets the charge carriers in the conductor in motion and thus leads to the induction voltage. Secondly: If a part of the conductor loop moves with the speed v in the magnetic field, the charge carriers are set in motion by the Lorentz force F = v x B, which also leads to a voltage. If the magnitude or direction of B is changed, the first effect is effective; if the area of the conductor loop is changed, the second effect is relevant. The two effects mentioned above can be used to calculate the induced voltage between the ends of the conductor for any conductor geometry, including those that do not include a surface. This is done in the program. A homogeneous field is used as the magnetic field, the magnitude of which can be changed and which can be rotated. The conductor loop can be drawn in any shape and the shape can be changed during the calculation. The program calculates the induced voltage in real time by changing the field and by changing the shape of the loop. The force on the charge carriers is marked in color at each point of the conductor loop. The magnetic field component perpendicular to the conductor loop and the induced voltage are continuously displayed as functions of time.
Downloads
Operation
Left-click several points in the image on the left to create a conductor loop. If you want to move a point, place the mouse on the point and move it while holding down the left mouse button. You can delete a point by pressing the right mouse button on it. The calculation no longer needs to be started separately, it continuously calculates the induced voltage in the background. There are three ways to make changes that lead to the induction of a voltage. 1) Change the amount of the magnetic field. This can be done with the middle slider. The faster you move the slider, the greater the induced voltage. 2) Changing the direction of the magnetic field. The magnetic field can be rotated by turning the gray disc with the red arrow. To do this, click on the disk and rotate it by holding down the left mouse button. The rotation takes place around the y-axis so that the field points out of the screen (+z), points to the right (+x), points into the screen (-z) or points to the left (-x). The third way to induce a voltage is to change the loop geometry. To do this, click on one of the red dots with the left mouse button and move it while holding down the mouse button. The faster you move it, the greater the induced voltage.
To take a closer look at two typical cases, the magnetic field can rotate automatically at a constant angular velocity or its magnitude can be changed automatically at a constant rate. Both can be selected at the bottom right.
Representation
The conductor loop is colored green or blue to represent the local force on the charge carriers. A force on positive charge carriers in the direction of the conductor loop (direction from the first to the last point) is shown in blue and a force against the direction of the conductor loop is shown in green. The force on the negative electrons naturally acts in the opposite direction (blue against the direction and green in the direction of the conductor). Black means there is no force. The color scale slowly adapts to the current maximum and minimum values. For this reason, you sometimes have to wait a little after strong changes before you can recognize colors again. The z-component of the magnetic field strength (component perpendicular to the screen plane) is displayed at the top right. This is the relevant component for the induction in the flat conductor loop. The induced voltage Uind, which is present between the start and end of the conductor loop, is displayed in the image below. The time scale of the display automatically adapts to the computing speed of your computer so that all calculations and displays are possible in real time.
Note
If the field strength of a homogeneous magnetic field is changed, the shape of the induced electric field is not clearly defined. The problem can only be solved by determining how the magnetic field lines of the magnetic field close again in the outer space. This program assumes a cylindrically symmetrical field geometry with its center in the middle of the screen. This is indicated by the concentric circles in the background. The induced electric field, which is calculated using the formula red E = - dB/dt, then has the form E(r) =(y,-x,0)*dBz/dt. The magnitude of E increases linearly with the distance from the center. With closed conductor loops, the induced voltage is independent of the position of the conductor loop. With open conductor loops, especially with straight conductor sections, the magnitude and even the sign of the induced voltage depends on the position of the conductor. This phenomenon is physically correct and not just an artifact of the calculation. So don't be surprised if you get different results depending on where you place a piece of conductor.
Numerical realization
The induced voltage is calculated as an integral over the entire conductor loop:
The changes in B and the shape of the conductor loop are recorded live from your inputs. This is used to calculate the field strength E and the speed of the individual conductor elements. The shape of the conductor loop is interpolated between the interpolation points using a cubic spline. Between two interpolation points, 100 conductor elements ds are used to calculate the integral.