Model 1.

 

We consider electric field by two identical oppositely charged very thin disks in the plane (see Figure M.1). This is a model for your first experiment where two circles are drawn with conductive ink on special slightly conductive paper (see Figure M.2). The charges of these circles are produced by connecting them to the voltage source that maintains known potential difference  between the disks. This is a “2D” case – electric field lies in the plane of the disks. You may think of these disks as cross-sections of very long charged cylinders (due to translational symmetry electric field lies in the plane perpendicular to the axes of the cylinders).  Our goal is to find electric field  and potential  along the line connecting the centers of the disks.

 

1. Recall that in “2D” case (I.6) the magnitude of the electric field due to charged disk is inversely proportional to the distance from its center:
   .


2. Let us determine the value of the constant . Assume the disks have radii  and their centers lie on  axis at points  and . Consider electric field at some point on the axis between the circles(see Figure M.1). We assume the “upper” circle is at potential  while the “lower” circle is at 0 potential (connected to GND). Electric field at point  due to “upper” positively charged disk is . Electric field at point   due to “lower” negatively charged disk is . Electric fields  and  have the same direction therefore the net field at point  is:

   (M.1)

   The difference in electric potential between the disks:

   (M.2)

   Integrate (M.2) and express  in terms of , , . Remember,  is a known quantity.



3. Now we will derive the expression for the potential  on the line connecting disks (-axis in Figure M.1). We know that the potential of the lower disk is 0. We will integrate electric field (M.1) from the lower disk to the point :

   

    

   

   or

   (M.3)

   You already expressed  in terms of , ,  in part 2. Integrate (M.3) and express  in terms of , ,  and, of course, .

 

 

What to submit in Prelab:

1.  in terms of , ,  (with short derivation)

2.  in terms of , , ,  (with short derivation)

 

 

Model 2.

Now we will derive expression for the electric potential between two concentric circles with known potential difference. A small conducting disk of radius  at potential  is concentric with a larger circle with inner radius  at potential  (see Figure M.3). Our goal is to find expressions for  and  where  is the distance to the center of the circles.

1. Electric field in this setup is directed radially outward and, due to circular symmetry, is inversely proportional to the distance from the center:
   


2. We need to find coefficient  and express it in terms of , , and . Express the difference in electric potential between the circles through the line integral of electric field along radial line:
   

   (M.4)

   Integrate (M.4) and use  and  to express coefficient  in terms of , , and .



3. Potential  at distance  from the center:
   

   (M.5)

   You already expressed  in terms of , ,  in part 2. Integrate (M.5) and express  in terms of , ,  and, of course, .

 

Note: an alternative approach of obtaining the expression for the potential  is solving the Laplace's equation in cylindrical coordinates  with boundary condition  and .

 

 

What to submit in Prelab:

1.  in terms of , ,  (with a very short derivation)

2.  in terms of , , ,  (with short derivation)