University Electromagnetism: Electric field and potential of a capacitor upon filling with a dielectric material (connected or not to a battery)

1. Filling a capacitor with a dielectric: I. Complete filling © Frits F.M. de Mul

2. Filling a capacitor with dielectric Available: Flat capacitor : = surface area A , = distance of plates d , = no fill (  r  . Question: What will happen with Q ,  E, D,  V and C upon filling the capacitor with dielectric material (with  r  A d Assume: initially, plates are charged with +Q , -Q. +Q -Q

3. Analysis Case A : yes Case B : no Consequences : in A : voltage remains constant in B : charges remain constant. A d +Q -Q Important : Does the capacitor remain connected to the battery that loaded it?  V

4. Approach (1) Assumption : no “edge effects” : d << plate dimensions Consequences : no E -field leakage planar symmetry everywhere : Gauss’ Law is applicable A d +Q -Q

5. Approach (2) A d +Q -Q Relevant variables : Q ,  , E, D,  V and C Material constants :    r , A and d . Task : Establish the relations between the variables.

6. Approach (3) Relations between variables : Q f A d +Q -Q D E Material constants: D =    r  E  V

7. Calculations (1) => D =  f   D Take Gauss pill box, top/bottom area = dA , enclosed charge =  f ..dA dA

8. Calculations (2)   E Take integration path from bottom to top, length d =>  V = E.d d

9. Calculations (3) D =  f = Q f /A D =    r E  V = E.d C = Q f /  V Case A : capacitor connected to battery  V remains constant !! Where to start????? Q f D E  V Q f  f  D E  V C  b = begin = end 1.  V = const start 2. E = const 3. D :  r x as large 4.  f :  r x as large 5. Q f :  r x as large 6. C :  r x as large 7.  b = (  r -1)  f, “old”

10. Calculations (4) Q f D E  V D =  f = Q f /A D =    r E  V = E.d C = Q f /  V Case B : capacitor NOT connected to battery Q f remains constant !! Q f  f  D E  V C  b Where to start????? 1. Q f = const start 2.  f = const 3. D = const 4. E :  r x as small 5.  V :  r x as small 6. C :  r x as large 7.  b = (1-  r )  f, “old”