Capacitors and Current

Dr Jacob Dunningham,School of Physics and Astronomy, University of Leeds

EM-L5-1

Review of Lecture 4

• Electric static energy

U =∫ Q

0V (q) dq

• Capacitance

C = Q/V ; U =1

2

1

CQ2

• Electric field energy density

ue =1

2ε0 E2

Review of Lecture 4 EM-L5-2

Overview of Lecture 5

The plan for today’s lecture

• Combinations of capacitors

• Dielectrics

• Electric current

• Summary

Review of Lecture 4 EM-L5-3

Combination of Capacitors

EM-L5-4

Parallel capacitors

Equal potential difference V across C1 and C2

Q = Q1 + Q2 = C1V + C2V = (C1 + C2) · V

Equivalent capacity

Ceq =Q

V= C1 + C2

Combination of Capacitors EM-L5-5

Parallel capacitors

In general for multiple parallel capacitors Ci:

C =∑i

Ci

Combination of Capacitors EM-L5-6

In series capacitors

Equal charges of ±Q on all plates

V = V1 + V2 =Q

C1+

Q

C2= Q ·

(1

C1+

1

C2

)Equivalent capacitance

1

C=

V

Q=

(1

C1+

1

C2

)

Combination of Capacitors EM-L5-7

In series capacitors

In general for capacitors Ci in series

1

C=∑i

1

Ci

Combination of Capacitors EM-L5-7

Example: equivalent capacity

Find the equivalent capacitance of the network of capacitors

Combination of Capacitors EM-L5-8

Example: equivalent capacity

Find the equivalent capacitance of the network of capacitors

Answer: Ctotal = 2 µF

Combination of Capacitors EM-L5-9

Dielectrics

EM-L5-10

Dielectric

• Definition: Dielectrics are non-conducting materials, such as

air, glass, paper, . . .

• Effect of a dielectric: A dielectric decreases the strength of

the electric field E relative to the electric field E0 in vacuum.

E = E0/κ

κ is called the dielectric constant.

Material dielectric dielectric

constant κ strength kV/mm

Air 1.00059 3

Glass (Pyrex) 5.6 14

Mica 5.4 10 - 100

Paper 3.7 16

Porcelain 7 5.7

Too strong a field can damage the dielectric.

Dielectrics EM-L5-11

Parallel plate capacitor dielectric

Potential difference between plates with separation d

V = E · d =E0 · d

κ=

V0

κ

Effect on capacitance C

C =Q

V=

Q

V0/κ= κ

Q

V0= κ C0

Capacitance becomes

C = κ C0 = κε0 A

d=

ε A

d

where the permittivity of the dielectric ε is

ε = κ · ε0

V0 potential without a dielectric. C0 capacitance without a dielectric.

Dielectrics EM-L5-12

Molecular level view

In an external electric field the dipoles become partially aligned.

Bound charges appear on the surface of the dielectric.

Dielectrics EM-L5-13

Bound charges reduce electric field

The bound charges on the surface of the dielectric generate an

electric field.

This additional field with opposite orientation reduces the exter-

nal electrical field strength.

Dielectrics EM-L5-14

Electric Current

EM-L5-15

Electric current

The electric current I is the amount of charge ∆Q thatflows through a cross sectional area A in time ∆t

Definition of electric current I

I =∆Q

∆t

Unit of current Ampere (A)

1 A =1 C

1 s

Electric Current EM-L5-16

Current and drift velocity

In time ∆t all N charges q in the shaded volume V pass throughthe surface A. With number density of charges, n = N/V :

∆Q = q · n · (A · vd ·∆t)

the current then is

I =∆Q

∆t

I = q · n · A · vd

(n - number density, n = N/V , A - cross sectional area, vd - drift velocity )

Electric Current EM-L5-17

Resistance

Current is driven by the electric field ~E inside the conductorelement of length ∆L exerting a force q · ~E on free charges. Thepotential difference is V = E ·∆L.

Definition of resistance

R(I) = V (I) / I

SI unit 1 Ohm Ω = 1 V/1 A

Ohms law: for many materials R is constant

R = V / I

Electric Current EM-L5-18

Example

(a) carbon or metal film resistor, (b) semiconductor diode

The resistance is not necessarily constant.

Electric Current EM-L5-19

Resistivity

Resistance of a wire depends on length L and cross section A.

R = ρ ·L

A

The resistivity ρ is a characteristic of the conducting material

Electric Current EM-L5-20

Summary

• Parallel capacitor

C =∑

i

Ci

• In series capacitors1

C=∑

i

1

Ci

• Dielectric

E=E0 / κ ; ε=κ · ε0

• Current

I = Q / t ; I = q · n · A · vd

• Resistance

R = V / I

Reading: Tipler, sections 24-4, 24-5, 24-6, 25-1, 25-2

Preparation: Tipler, sections 25-3, 25-4, 25-5

Summary EM-L5-21