Capacitors

1. Capacitors with Dielectrics
1

2. Dielectric
•A dielectric is a nonconducting material inserted between
the plates of a capacitor.
•A dielectric increases the ability of a capacitor to store
energy.
•If the dielectric completely fills the space between the
plates, the capacitance increases by a factor k, called the
dielectric constant.
C 
k εo A
d
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3. Dielectric
•When a dielectric is
inserted between the
plates of a charged
capacitor that is not
connected to a battery (a
source of additional
charge), but the voltage is
reduced by a factor k.
k 
Vold
Vnew
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4. Dielectric
•Since the charge Q on the capacitor does not change,
then the capacitance must change with the changing
voltage so that the charge remains constant.
k 
Cnew
Cold
•The capacitance increases by a factor k when the
dielectric completely fills the space between the
plates.
•The dielectric constant is a measure of the degree of
dipole alignment in the material.
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5. d
the capacitance can be increased by decreasing the
distance d between the plates.
•The value of d is limited by the electrical discharge that
could occur through the dielectric medium separating the
plates.
•For any separation d, the maximum voltage that can be
applied across the capacitor plates without causing a
discharge depends on the dielectric strength of the
dielectric.
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•From the equation for capacitance
C 
k εo A

6. •In other words, the
maximum electric field the
dielectric material can
withstand without allowing
a transfer of charge
between the plates is the
dielectric strength.
•If the electric field strength
in the dielectric exceeds the
dielectric strength, the
insulating properties will
break down and the
dielectric material begins to
conduct. This is called
dielectric breakdown.
d
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V
dielectric) 

ds(E

7. •“Polarization” of a dielectric in an
electric field E gives rise to thin
layers of bound charges on the
dielectric’s surfaces, creating
surface charge densities +i and –
i.
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8. An Atomic Description of Dielectrics
(a) Polar molecules are randomly oriented in the
absence of an external electric field.
(b) When an external field is applied (to the right as
shown), the molecules partially align with the field;
the dielectric is now polarized.
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9. (a) When a dielectric is polarized, the dipole moments of the
molecules in the dielectric are partially aligned with the
external field Eo. (b) This polarization causes an induced
charge on the opposite side. This separation of charge results
in a reduction in the net electric field within the dielectric.
The net effect on the dielectric is the formation of an induced
positive surface charge density sind on the right face and an
equal negative surface charge density –sind on the left face.
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10. The induced surface charge give rise to
an induced electric field Eind in the
direction opposite the external field Eo.
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11. What is the Magnitude of the Induced Charge Density?
For parallel plate capacitor,
External field Eo = /o
Induced Field Eind = ind/o
and E = Eo/k = /k·o
Substitute into E = Eo - Eind gives
Note that the induced
charge density on the
dielectric is less than the
charge density on the
plates.

 
ind ;
ind 
 
o o o o
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k
o k
o
ind
k


o 


o 
 

ind 
o ko k
k 
 k 1
k k
 

 
 

12. Dielectrics
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13. E n e w
•The dielectric provides these advantages:
•Increases the capacitance of the capacitor.
•Increases the maximum operating voltage of a capacitor.
•May provide mechanical support between the
conducting plates.
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E o ld
• The net electric field in the dielectric is given by: E = Eo - Einduced

k  

14. Effect of the Dielectric Constant

Eo
dielectric
E
k
Vo
dielectric
k

Eo
E d d
Vdielectric 
k
Potential difference with a dielectric
is less than the potential difference
across free space
o
o
V V
C 
Q
 k 
Q
 k C Results in a higher capacitance.
Allows more charge to be stored before breakdown
Neo
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tec
o
h In
ls
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tita
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of T
e
ec.
hnology 14

15. Effect of the Dielectric Constant
Parallel Plate Capacitor Co 
Material permittivity measures
degree to which the material
permits induced dipoles
to align with an external field.
oA k o A
 C  k Co 
d d
o
 k 
C 
A
d
E2
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o 0
 E2
 u  0
k  E2

1
2
1
2
1
2
u 
Example modifications
using permittivity

16. •The energy of the capacitor is lowered
when a dielectric is inserted between the
plates. Work is done on the dielectric.
•A force must act on the dielectric which
pulls it into the capacitor.
•The nonuniform electric field near the
edges of a parallel plate capacitor exerts
this force.
•The horizontal component of the electric
field acts on the induced charges on the
surface of the dielectric, producing a
horizontal force directed into the capacitor.
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17. •Place a dielectric material between
capacitor plates:
• A charge of -13 is on the right negative
plate of the capacitor.
• A charge of +7 is on the right positive
portion of the dielectric.
• The battery sees -13 + 7 = -6 (less total
charge on the negative capacitor plate).
• To compensate for the absence of charge,
the battery sends more charge to the
negative capacitor plate to restore the -13
charge.
• This is how a dielectric allows for more
charge to be stored on the capacitor
plates.
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18. Effect of a Metallic Slab Between the Plates
A parallel-plate capacitor has a plate separation d and
plate area A. An uncharged metallic slab of thickness a is
inserted midway between the plates.
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19. Charge on one plate must induce a charge of equal magnitude but
opposite sign on the near side of the metallic slab.
Net charge on the slab is zero, electric field inside the slab is zero.
Capacitor = two capacitor in series, each having a plate separation of
(d-a)/2 as shown. Neotech Institute of Technology 19

20. 2
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1

1

1
;
1

1
C
T C
1 C2 C
T 
o 
A
da
2
1
1

o 
A
da
2
2

o 
A
2
1 da

o 
A
CT
1

2

o 
A
da
CT
;
1

CT
;
1

o
T
; C 
da da
 
A
da


C
T 
o 
A

21. Effect of a Metallic Slab between the plates
What are the differences in the result as compared to the
previous example if we insert the metallic slab as shown?
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22. A Partially Filled Capacitor
A parallel-plate capacitor with a plate separation d has
a capacitance Co in the absence of a dielectric. What is
the capacitance when a slab of dielectric material of
dielectric constant k and thickness d/3 is inserted
between the plates as shown.
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23. A Partially Filled Capacitor
Two capacitors in
series
1
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2
3 3
1
3
3
1
1
2 d
2 d
3 o A
1
2 k d
3 k o A

3 k o A
CT 3 k o A d (1 2 k)
CT
CT
T
1

d  2 k d
;C
d

k o A
;C 
o A
2 d
d
1

1
CT C1 C2
1

1
;C
d
d
 
1
k o A o A
2 d

1
3 k o A 3 o A


CT 3 k o A
1 d


CT 3 k o A


24. Two Dielectric Slabs
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• Consider a parallel-plate
capacitor that has the space
between the plates filled with
two dielectric slabs, one with
constant k1 and one with
constant k2.
• The thickness of each slab is
the same as the plate
separation d and each slab
fills half the volume between
the plates.
• The two different dielectric
slabs represent two
capacitors in parallel with a
plate area A/2.

25. Two Dielectric Slabs
CT  C1  C2
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
k1 o A

k2 o
2 d 2 d

k1 o A  k2 o A
2 d

o A (k1  k2 )
2 d
T
T
T
A
C
C
C