Question

1. Consider placing two point charges on an x-y plane, the first charge, q1, at (x1, y1), and the second charge , q2, at (x2, y2).

2. Derive an expression for the electric potential and field at any arbitrary point (x, y) in terms of q1, q2, x1, x2, y1 and y2.

3. Choose some reasonable values for q1, q2, x1, x2, y1 and y2 and make a rough sketch of what you expect the electric potential/field to look like. Your picture doesn’t necessarily have to be correct, but you should include some justification for why your sketch looks the way it does. 4. Use SageMath to draw the exact electric potential and field using the expression derived above. Below is an example. x, y = var("x y") g = Graphics() g += contour_plot(1.5 + 0.2*x*y, (x, -4, 4), (y, -4, 4), fill=False, cmap="jet", labels=True, contours=[0, 1, 2, 3, 4], label_fontsize=14) g += plot_vector_field((y/2, -x/2) , (x, -4, 4), (y, -4, 4)) g.show() Your final result should look something like the figure below. Note how you can clearly see the two point charges at (1, 2) and (2, 1.5).

Answer 1

2.

Electric potential:

The potential due to charge q1(x1,y1) at (x,y) :

Similarly, for q2:

Therefore, the total potential at (x,y)

3.

Let us take

Also, let's take (in magnitude)

Let

and

So, we are left with:

For isolines (equipotential) lines,

we set the above expression to some constant.

Since there are squares involved, the isolines would be closed curves.

If we set y=0,

Since this equation has a form of: , it has a parabolic nature when , similarly for y, it will have a parabolic nature when .

When these to natures come together to form a closed curve, it will be elliptical with the charges at the foci.

4.

The vector field is actually the electric field.

The electric field is:

Here's the code:

x, y = var("x y")
f = Graphics()
f += contour_plot(1/(sqrt((x-1)^2+y^2))+1/(sqrt((y-1)^2+x^2)), (x, -2, 3), (y, -2, 3), fill=False, cmap="jet", labels=True, contours=[0, 0.5,1,1.5, 2,2.5, 3,3.5, 4], label_fontsize=14)
f += plot_vector_field(( (x-1)/((x-1)^2+y^2)^1.5+x/(x^2+(y-1)^2)^1.5,y/((x-1)^2+y^2)^1.5+(y-1)/(x^2+(y-1)^2)^1.5 ) , (x, -2, 3), (y, -2, 3))
f.show()