Quiz 4
Phys 3310

Two point charges, q and -q, are distance 'a' apart from each other and distance 'a/2'  apart form a conducting, grounded infinite plane.
a) What is the net force on the positive charge? 
b) What is the electrostatic energy of this configuration?

                               a
                         <------------------------->
                 -q   *                                                           * q
                         |     
             a/2       |
    _______ _  _|_____________________________________
                  conducting grounded plane

Solution:

The equivalent image charge problem  is 
                              a
                       <------------------------------>
                 -q   *                                                                   * q
                      |     
            a/2     |
    . . . . . . . . . | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
                      |
                      |
                      |  
                q   *                                                                      *  -q      
                
                The force
                \(TraditionalForm\`F\_x = \(\(q\/\(4  πϵ\_0\)\) \((\(-q\)\/a + \(q\/\(\(\@2\) 
                        a\)\) \@2\/2)\) = \(-\(q\^2\/\(8 \( πϵ\_0\) a\)\)\)\)\)
                (Subscript[F, y])= \(TraditionalForm\`\(q\/\(4  πϵ\_0\)\) \((\(\(-q\)\/a\)\(+\)\)\)  \(TraditionalForm\`\(\(\(q\/\(\(\@2\) 
                  a\)\) \@2\/2\)\()\)\) = \(-\(q\^2\/\(8 \( πϵ\_0\) a\)\)\)\) 
                
                The components are negative because the force points in the negative direction both in x and y.
                The energy is half of the image charge energy 
                W= (1/2) ((1/2) Σ(Subscript[q, i]) (Subscript[V, i]))
                Since (Subscript[q, i]) (Subscript[V, i])  is the same for all 4 charges
                W= \(TraditionalForm\`\(\(\(1\/2\) \((\(1\/2\) \((\(q\/\(4  πϵ\_0\)\) \((\(q\/a\) 2 - q\/\(\(\@2\) a\))\))\))\) 
          4 = \(-\(\(\(\ \)\(\((\(-4\) + \@2)\)\ q\^2\)\)\/\(8\ a\ πϵ\_0\)\)\)\)\(
\)
      \)\)