**Figure 1**: Current density induced on copper disc in A/mm².

$$∇×E↖{→}+∂B↖{→}/∂t=0,$$

where $E↖{→}$ denotes the electric and $B↖{→}$ the magnetic field. Gauss's law $∇E↖{→}=ρ/ε_0$, links the electric field electric charge $p$ via the permitivity of free space $ε_0$.

In the absence of electric charge, a change in the magnetic field is the only source for the electric field. The governing equations are given by:

$$∇×E↖{→}+∂B↖{→}/∂t=0, ∇⋅E↖{→}=0.$$

Mathematically this is identical to magnetostatics,

$$∇×B↖{→}-μ_0 J↖{→}=0, ∇⋅B↖{→}=0,$$

where $μ_0$ denotes the permeability of free space.

As such, Faraday-induced electric fields - that is fields generated solely by a change in the magnetic field - are determined by $-∂B↖{→}/∂t$ in the same way that magnetostatic fields are determined by $μ_0 J↖{→}$. The Maxwell–Faraday law in integral form is therefore closely related to Ampère’s law:

$$∮_∂Σ{E↖{→}dl↖{→}}=-d/{dt} ∬_ΣB↖{→} {da↖{→}}=-{dΦ}/{dt}$$

(Maxwell–Faraday law in integral form)

$$∮_∂ΣB↖{→}dl↖{→}=μ_0 ∬_ΣJ↖{→} {da↖{→}}=μ_0 I_{enclosed}.$$

(Ampère’s law from magnetostatics)

Furthermore, if an electric field is induced in a conductor this generates a current, as shown by Ohm’s law, $ J↖{→}=σE↖{→}$, where $J↖{→}$ denotes the current field and the conductivity. It follows that, if a conductor is placed within a region of space where we have a magnetic field that changes over time, a current will be induced as well.

For a time harmonic magnetic field $B↖{→}(t)=B_z e^{jωt} e↖{→}_z$, where $ω=2πν$ denotes the angular frequency, the time derivative is given by $∂/∂t B↖{→}(t)=jωB↖{→}(t)$. For many metals the permeability $μ$ is very close to the permeability of free space $μ_0$, such that effects from magnetization can be neglected with regard to the field.

For a circular, conducting surface orthogonal to $e↖{→}_z$ we can exploit the symmetry to solve for the electric field using the Maxwell–Faraday law (the same basic calculation can be found in [1]):

$∮E↖{→}(t)dl↖{→}=2πrE(t)=-πr^2 jωB(t)=-d/{dt} (πr^2 B(t))=-d/{dt} ∬B↖{→}(t)da↖{→}$ |
(1) |

$E↖{→}(t)=-1/2 jωrB(t) e↖{→}_φ.$ |
(2) |

Fig. 2 shows a schematic representation of the symmetry involved.

By using Ohm’s law one immediately arrives at the current distribution,

$J↖{→}=σE↖{→}=-1/2 σjωrB(t) e↖{→}_φ,$ |
(3) |

or

$J↖{→}=-1/2 σjωrμ_0 H(t) e↖{→}_φ=-jσπνrμ_0 H(t) e↖{→}_φ,$ |
(4) |

if we neglect the magnetic susceptibility of the material. In the case of copper, with $μ_r≈1$ this is a justified approach.

**Question**

What direction does the current have when the magnetic field decreases in *z*-direction / increases in *z*-direction?

The current will in turn induce a magnetic field. What is the qualitative shape of this magnetic field? As the current distribution changes so will the field induced by the current, which in turn influences the current. Why can we neglect this in our consideration?

A circular copper disk of radius 2 mm and depth 0.5 mm is defined. Additionally we define a time harmonic homogeneous magnetic source field which is oriented orthogonally to the circular surface of the disc. In accordance with the source field, the boundary conditions in the *z*-direction are set to ‘magnetic’. With the cylindrical disc it is possible to activate symmetry conditions which help to reduce mesh size and therefore calculation time. The basic setup is shown in fig. 3.

Parameter | Value | Description |

Freq | 50 Hz | Evaluation frequency |

Hz | 1e03 | Amplitude factor |

By setting up a curve along the radius of the disc we can directly evaluate the induced current distribution in the disc and compare the results with the analytical expression. For this purpose the corresponding post-processing templates are used.

**Figure 3**: CST problem setup of a copper disc in a homogeneous time harmonic magnetic field. Left hand side: copper disc and bounding box. Right hand side: copper disc set against the time harmonic source field. The source field is depicted in the main cross sections of the disc along the z-axis.

The following links allow you a closer look at the model construction and give you the opportunity to download the ready-made CST-files for both the student and the full version.

Model Construction Watch Video

Download Model File (Student Edition) Download Model File

To compare analytical and numerical results go to the “Post Processing” tab and click on “Template Based Post Processing” as shown in fig. 4 a).

**Figure 4**:** a)** “Template Based Post Processing” button in the “Post Processing” tab.** b)** “Evaluate Field on Curve” option.

The “Template Based Postprocessing” window offers a variety of features. To evaluate the field on a curve choose “2D and 3D Field Results” in the first drop down menu and “Evaluate Field on Curve” in the second. Browse for results, select the Conduction Current Density, set “Component” to “Abs” and “Complex” to “Mag” to get the magnitude of the conduction current (the main steps are shown in fig. 4). The default name for the result is “curve1_Cond. Current Dens. (Freq)”, indicating that the conduction current density is evaluated along curve1 at frequency “Freq”. However, barring a few special characters, you can enter an arbitrary result name. When comparing simulation results with analytical results it might e.g. make sense to use “Simulation” instead of “curve1_Cond. Current Dens. (Freq)”. To evaluate the template click on the “Evaluate” button in the “Template Based Postprocessing” window.

As we can see from eq. (4) we need the conductivity of the disc in order to evaluate the analytical expression for the conduction current density. To find the respective value open the material folder in the Navigation Tree (look to the left hand side) and double-click on *Copper (annealed)* (cf. fig. 5 a)) to open the Material Parameters dialog. You will find the conductivity value in the “Conductivity” tab (cf. fig. 5 b)). Next, set up a new parameter named “conductivity” in the Parameter List and enter the value in the “Expression” column. To set up the parameter simply double-click on “<new parameter>” in the Parameter List on the bottom of the CST window (cf. fig. 5 c)).

**Figure 5**: **a)** In the Materials folder in the Navigation Tree double-click on “Copper (annealed)”. **b)** The Material Parameters dialog. **c)** To enter a new parameter in the Parameter List simply double-click on <new parameter>.

There are several ways to plot the analytical expression. In this example we will use the option “0D or 1D Result from 1D Result (Rescale, Derivation, etc.)”. To do so, select “General 1D” in the first and “0D or 1D Result from 1D Result (Rescale, Derivation, etc.)” in the second drop down menu. Fig. 6 shows the resulting dialog and required input. First, mark the “1D(C)” (upper left corner of the “Specify Action”-frame), indicating that you are interested in generating a continuous result. “Axes: Scale with Function f(x,y)” should already be preselected. In the “1D Result”-frame select the previous template. In figure 6 the name is given as “Simulation”, the default would be “curve1_Cond. Current Dens. (Freq)”. Back in the “Specify Action”-frame choose “y-axis” in the “Apply to” field. This indicates that we will be using the *x*-axis from the selected result and scaling the *y*-axis of the plot. In the “f(x,y)”-field enter the expression we have derived above in eq. (2). You can simply copy and paste the formula from the caption of fig. 6, but if you used different parameters, you will have to modify the formula accordingly. The constants “pi” and “mue0” do not need to be defined separately, as they are part of a set of predefined values, meant to ease calculations.

**Figure 6**: “0D or 1D Result from 1D Result (Rescale, Derivation, etc)” window. Select the simulation result and scale the ordinate to reflect the analytical result.

**Analytical expression: **conductivity * pi * Freq * x * 1.0e-3 * mue0 * Hz

Keep in mind that we are using mm as length unit. Therefore a factor of $10^{-3}$ has been added to the expression.

So far the templates plot the field results vs. the curve coordinates, i.e. the length of the curve. This can be bothersome if the curve was set up not from the center outward, but from the rim to the center. In this case you can either set up a new curve, or e.g. replot with the axis coordinates. To do the latter a second post processing task has to be performed. Select “General 1D” in the first drop down menu and “0D or 1D Result from 1D Result (Rescale, Derivation, etc)” in the second. In the “0D or 1D Result from 1D Result” dialog check “1D(C)” and select “Parametric X-Y Plot”. Then select the *x*-coordinates of the curve as x-values (Field Along Curves\curve1\Coordinates\X) and the results for the *y*-axis as shown in fig. 7.

**Figure 7**: Initiate a new post processing task to plot your results vs. the actual coordinates on the curve.

Click on “Evaluate All” in the “Template Based Postprocessing” window to evaluate all post processing steps. The corresponding results can be found in the navigation tree under “Tables\1D Results\”. Select the whole folder in order to show all contents or make a subselection in order to show only specific results.

You can change axis labels and plot title, but in order to so, first you must establish a new subfolder, e.g. named “Compare Results”, in “1D Results” in the navigation tree and copy the results there. Fig. 8 shows the comparison between the simulation results and the analytical expression for the conduction current. Axis labels and title can be changed in the “Home” tab using the “Properties” button in the “Edit” section.

**Figure 8**: Comparison of results for the conduction current density along the radius of the copper disc.

As we can see the analytical and numerical results show a very good agreement.

What direction does the current have when the magnetic field decreases in the *z*-direction / increases in the *z*-direction?

The current will in turn induce a magnetic field. What is the qualitative shape of this magnetic field? As the current distribution changes so will the field induced by the current, which in turn influences the current. Why can we neglect this in our consideration?

[1] = David J. Griffiths , Introduction to Electrodynamics, fourth edition, Example 7.7, p. 317

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