1. PHYS219 Fall semester 2014 Lecture 06: Capacitors Dimitrios Giannios
Purdue University

2. The standard simple modelΔ
E inside is uniform to a good approximation.
E outside is zero to a good approximation
parallel plate capacitor filled with air
-12
Note 1: εo=8.85×10 C2/(N•m2)
Note 2: C depends on geometry
Note 3: A is area of one plate 2

3. Relating V to E – general case
Δ(PEE) + W=0 F
F =qtest E
F|| = qtest E||
initial q
test Δs
ΔW= F|| Δs
W = ΣΔW = qtest ΣE|| Δs
= -Δ(PEE) = -qtestΔV Q4 Q1 x
final Q Q2 3
How V changes with Δ s gives quantitative
information about how E varies with position

4. How is the potential difference ΔV related to E for the case of uniform E field?
Two parallel charged plates separated by a distance d
(the case of uniform E)
FE = qtest E Δs
W = FE Δs = qtest E Δs
but W = qtest ΔV
Δ V = E Δ s or
(previous slide)
The variation in electric potential with position is linked to the electric field!

5. Example
Increasing electrostatic potential
What is the magnitude and direction of the uniform electric field if the potential V is measured to be
V1 = +15 V and V2 = +10 V
at two points separated by 0.2 m? +
____ _
_ _ _ _ x

6. Example: Variation of potential in 1-dimension

7. Derivation for Capacitance of Parallel Plates
Focus on positive plate
First, let’s calculate the electric field

8. Derivation for Capacitance of Parallel Plates
Focus on positive plate
Gaussian surface
qenclosed
Gauss ' Law: E ,total  ε0

9. Derivation for Capacitance of Parallel Plates (More)
q is charge on ONE plate Enet is field inside capacitor
Negative plate
Positive plate
Gaussian surface
d (separation between plates)
ΔV (potential difference between plates) E
neg
In general, we also have

10. Nature’s capacitors
Although relatively simple, the parallel plate capacitor can be used to provide simple models for capacitance in many situations, for example C1 C2

11. Energy Stored in Capacitors
Example: If C=1μF and ΔV=1.5 V, how much energy is stored?

12. Real life capacitors
Most real capacitors contain two metal “plates” separated by a thin insulating region
Many times these plates are rolled into cylinders
The region between the plates typically contains a material called a dielectric

13. The effect of dielectrics on capacitance (qualitative)11
Uncharged capacitor
for given charge at capacitor
 Electric field reduced
voltage reduced
Capacitance increased Δ

14. κ = The effect of dielectrics (quantitative) Evacuum
dielectric constant
κ =
Edielectric

15. Manipulating capacitance:
Capacitors can be connected in one of two ways
Series: +Q ‐Q
=Q/(ΔV1+ΔV2) +Q ‐Q
Parallel:
=(Q1+Q2)/ΔV

16. What is the equivalent capacitance of this network?
Example:
What is the equivalent capacitance of this network?
C/4

17. Example II: What is the equivalent capacitance? this network?
15.33 μF
3.33 μF
Step 2:
Step 1:
Cequiv = C1 +C2 + C3 =
= 15.33