Molly Ayala
In order to plot the electric potential gradient using COMSOL, you must first understand the relationship between the electric field and the gradient of the electric field. In COMSOL notation, the electric field is represented as Ex, Ey, and Ez, while the gradient of the electric field is denoted as d(Ex,x), d(Ex,y), and d(Ex,z). To plot the electric potential gradient, you can utilize the Results feature in your COMSOL model. By right-clicking on Results, you can create a 1D Plot group and then add a Line graph by right-clicking again. This allows you to select the boundary and quantity of interest. For a 2D system, you can disable the surface plot and plot the arrow surface to visualize the electric field. By modifying the settings, you can plot the gradient of the electric field by changing the x and y components. Similarly, for a 3D system, you can use the arrow volume feature to plot the electric field and adjust the x, y, and z components to plot the gradient.
| Characteristics | Values |
|---|---|
| Software | COMSOL Multiphysics® |
| Version | 4.1 |
| Plot type | 1D Plot group, Line graph |
| Plot settings | Arrow surface, 2D Cut Line, 3D Cut Line |
| Electric field components | Ex, Ey, Ez |
| Gradient components | Vx, Vy |
| Equation | diff(normE,x) |
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What You'll Learn
Plotting electric potential in 2D
To plot the electric potential in 2D, you will need to use a specialised software such as MATLAB, Mathematica, or COMSOL Multiphysics®. The process will involve setting up the physical equations in vectorised form and calculating them for a grid of coordinates.
First, ensure you have the correct equation for the electric potential in the form of V = k * Q(ii) ./ hypot(p(ii,1)-X, p(ii,2)-Y). Then, follow these steps:
- Create a grid of coordinates where V is to be calculated: [X,Y] = meshgrid(0:0.05:4).
- Start with zero electric potential: V = zeros(size(X)).
- Superpose the electric potential field of each charge: for ii = 1:numel(Q) % Superpose the electric potential field of each charge.
- Calculate V using the equation: V = V + k * Q(ii) ./ hypot(p(ii,1)-X, p(ii,2)-Y).
- Plot the data: hContour = contourf(X,Y,V).
- Add a colour bar: hColorbar = colourbar.
- Label the y-axis: ylabel(hColorbar,'Electric potential (V)').
Note that the default contour spacing will be very tight around the point charges due to the singularities in the electrical potential they create.
For a 2D system in COMSOL, you will need to use the electrostatics module. Design your system and assign an electric potential for one electrode and ground for the other. When you run the study, you will automatically get a "surface" plot. Click on "arrow surface" at the top of your screen and disable the "surface" plot. Plot the "arrow surface". Change the "es.Ex" and "es.Ey" in the settings tab to the correct values.
In Mathematica, the process is similar, but you will need to convert polar coordinates to Cartesian coordinates (x and y) before plotting. You can use the gradientFieldPlot function to plot the electric potential and field together, with colours representing field strength and orange lines indicating field lines.
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Plotting electric potential in 3D
N = 1;
V0 = 1;
A = 1;
% or whatever your V0 or a are supposed to be
X = -1:0.01:1;
% this specifies x from -1 to +1 in 0.01 step size
Y = -1:0.01:1;
% this specifies y from -1 to +1 in 0.01 step size
[xx, yy] = meshgrid(x,y);
V = 4*V0/pi*1/n*exp(-n*pi*xx/a).*sin(n*pi*yy/a);
Surf(x,y,V)
This code defines the variables n, V0, and a, and specifies the range of x and y values. It then creates a meshgrid and calculates the electric potential V using the given formula. Finally, it plots the electric potential using the "surf" function.
Another software tool that can be used for plotting electric potential in 3D is COMSOL Multiphysics®. While discussing an older version of the software, one user mentions that they were working on a 3D model of a resistive current sensor and wanted to verify the high-frequency response. They encountered some challenges with erroneous results when using boundary probes to measure voltage potentials.
In addition to MATLAB and COMSOL, other software tools mentioned for plotting electric potential in 3D include Microsoft Mathematics, Scilab, and Python. However, some users have reported encountering difficulties when using these tools for this specific task. For example, one user had trouble plotting a 3D contour of electric potential in Python and sought help on Stack Overflow.
Overall, plotting electric potential in 3D requires the use of specific software tools and a good understanding of the underlying mathematics and equations involved.
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Calculating the gradient of an electric field
The gradient of an electric field is a fundamental concept in physics, particularly in the study of atomic, molecular, and solid-state systems. It measures the rate of change of the electric field at a specific point, often near an atomic nucleus. This rate of change is influenced by the distribution of electronic charges and the presence of other atomic nuclei.
In mathematical terms, the gradient of a scalar field (such as electric potential) is a vector that indicates the direction in which the field increases most rapidly. In the context of electric fields, the gradient of the electric potential at a point, after a change of sign, gives the electric field intensity at that point. This relationship is described by Equation 5.14.8.
When working with specific software tools like COMSOL, the calculation of electric field gradients can be more nuanced. In one instance, a user working with a 3D model of a resistive current sensor encountered a discrepancy between voltage (V) and electric field (E) values. They observed that the voltage potential field did not align with their expectations, despite the electric field corresponding well. This issue was attributed to the model's geometry and the interaction with the rapidly changing magnetic flux.
To calculate the gradient of the electric field in COMSOL, one can use the EC/ES physics notation: Ex_... = -dV/dx, Ey_... = -dV/dy, and Ez_... = -dV/dz. To obtain the electric field gradient, a further derivative is required: Exx_... = d(Ex_...,x), Exy_... = d(Ex_...,y), and similarly for other components. For plotting the gradient in 2D, the arrow surface option is used, and the "es.Ex" and "es.Ey" values are modified to their respective derivatives. In 3D, the equivalent option is "arrow volume", and the x, y, and z components are adjusted accordingly.
It is important to note that the electric field is the negative of the potential gradient, as represented by E = -Vx in COMSOL. Additionally, the diff() function can be used to calculate derivatives, such as diff(normE,x) for the x-derivative of the electric field's norm.
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Plotting the gradient of the electric field squared
To plot the gradient of the electric field squared, you need to follow a few steps, and the process differs for 2D and 3D systems. Here is a step-by-step guide for both scenarios:
2D System:
- Ensure you are using the electrostatics module and that you know how to design your system.
- Assign an electric potential for one electrode and assign the ground for the other electrode.
- When you run the study, you will initially get a "surface" plot. Click on the "arrow surface" option at the top of your screen.
- Disable the "surface" plot and plot the "arrow surface". This will give you the electric field, not its gradient.
- In the "settings" tab, edit the "es.Ex" and "es.Ey" fields. Change "es.Ex" to "d((es.Ex)^2+(es.Ey)^2,x)" and "es.Ey" to "d((es.Ex)^2+(es.Ey)^2,y)".
- Click "plot". The result may not be as expected, and you may need to make further adjustments.
3D System:
- Similar to the 2D case, first ensure you are using the electrostatics module and have designed your system appropriately.
- Assign electric potentials and ground as required.
- For a 3D system, you will be looking for the "arrow volume" option, which will provide a 3D plot of the electric field.
- Choose normalized arrows for the electric field to make it visually closer to what you might expect.
- For the gradient, change the x, y, and z components. Change "es.Ez" to "d((es.Ex)^2+(es.Ey)^2+(es.Ez)^2,z)".
In both cases, you are editing the "es.Ex" and "es.Ey" fields to calculate the gradient of the electric field. The process is similar for 3D, with the addition of editing the "es.Ez" field. The electric field is related to the potential gradient, and in Comsol, the electric field is the negative of the potential gradient.
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Using the Results section to plot electrical potential
To plot electrical potential using the Results section, follow these steps:
Firstly, right-click on "Results" and create a 1D Plot group. Then, right-click on the new 1D Plot group and add a "Line graph". You will then need to select the boundary and quantity of interest. If you want to plot along a random line in your geometry, go to "Results", right-click "Data sets", and select "2D Cut Line" or "3D Cut Line" for 3D geometries. Enter the coordinates for your line and click "Plot" to check its placement. Left-click on the 1D Plot group and select "Line plot 1" from the "Data set".
For 2D systems, you will get a "surface" plot automatically. Click on "Arrow surface" at the top of your screen and disable the "surface" plot. Plot the "arrow surface" and change the settings. Edit the "es.Ex" and "es.Ey" fields by adding the derivatives with respect to x and y, respectively. Click "Plot" to visualize the gradient of the electric field.
For 3D systems, the plot is called "arrow volume". Change the x, y, and z components of the electric field and then plot the results. You can also choose normalized arrows for the electric field to match your expectations.
In COMSOL, the electric field is denoted as -Vx, indicating the negative of the potential gradient. The gradient of potential with respect to x and y are "Vx" and "Vy", respectively. To obtain the gradient of the electric field, an additional derivative is required.
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Frequently asked questions
To plot the electric potential gradient in COMSOL, you need to first understand the gradient of the electric field. In Comsol notation for EC/ES physics, the electric field is the negative of the potential gradient, so Ex_... = -dV/dx, Ey_... = -dV/dy, and Ez_... = -dV/dz. To obtain the electric field gradient, you need one more derivative.
To plot the electric field gradient in 2D, you need to disable the "surface" plot and plot the "arrow surface". In the settings tab, change "es.Ex" to "d((es.Ex)^2+(es.Ey)^2,x)" and "es.Ey" to "d((es.Ex)^2+(es.Ey)^2,y)", then hit "plot".
To plot the electric potential profile along the x and y-axis, go to the Results part of your model. Make a 1D Plot group by right-clicking "Results", then add a "Line graph" by right-clicking the new 1D Plot group. Select the boundary and quantity of interest.
