Chapter 19

Electric Potential Energy and the

Electric Potential

19.5 Capacitors and Dielectrics

A parallel plate capacitor consists of two metal plates, one carrying charge +q and the other carrying charge –q. It is common to fill the region between the plates with an electrically insulating substance called a dielectric.

Advantages of dielectrics: •  they break down (spark) less easily than air •  the capacitor plates can be put closer together •  they increase the ability to store charge

19.5 Capacitors and Dielectrics

THE RELATION BETWEEN CHARGE AND POTENTIAL DIFFERENCE FOR A CAPACITOR The magnitude of the charge in each place of the capacitor is directly proportional to the magnitude of the potential difference between the plates.

CVq =The capacitance C is the proportionality constant. SI Unit of Capacitance: coulomb/volt = farad (F)

The larger C is, the more charge the capacitor can hold for a given V

19.5 Capacitors and Dielectrics

THE DIELECTRIC CONSTANT

If a dielectric is inserted between the plates of a capacitor, the capacitance can increase markedly.

Dielectric constant EEo=κ

E0

E E0

Some lines of force now terminate or start on induced surface charges E ≤ E0

Since E ≤ E0 --> κ ≥ 1

19.5 Capacitors and Dielectrics

19.5 Capacitors and Dielectrics

THE CAPACITANCE OF A PARALLEL PLATE CAPACITOR

dVE

E o ==κ

( )AqE oo ε=

VdA

q o !"

#$%

&=κε

dA

C oκε=Parallel plate capacitor

filled with a dielectric

<--> q = CV

C increases for --> increasing κ, increasing A, decreasing d

Example. How large would the plates have to be for a 1 F air-filled parallel plate capacitor with plate separation a) d=1 mm, b) d=1 µm ?

a)  d = 1 mm C = κε0A/d = ε0A/d (since κ = 1 for air) --> A = Cd/ε0 = (1)(.001)/(8.85 x 10-12) = 1.13 x 108 m2

If the plates were square, they would be of length and width (A)1/2 = (1.13 x 108)1/2 = 11 km --> very large!! b) d = 1µm = 10-6 m ~ the thickness of paper – use paper: κ = 3.3 --> A = Cd/(κε0) = (1)(10-6)/(3.3 x 8.85 x 10-12) = 3.42 x 104 m2 and (A)1/2 = (3.42 x 104)1/2 = 185 m --> about 2 football fields in length,

improving!

19.5 Capacitors and Dielectrics

Conceptual Example 11 The Effect of a Dielectric When a Capacitor Has a Constant Charge An empty capacitor is connected to a battery and charged up. The capacitor is then disconnected from the battery, and a slab of dielectric material is inserted between the plates. Does the voltage across the plates increase, remain the same, or decrease?

Effect of a Dielectric When a Capacitor Has a Constant Charge -- continued

Before inserting dielectric: q = C0 V0, C0 = ε0A/d

After inserting dielectric: q = C V, C = κε0A/d = κC0

C0 V0 = q = C V = κC0 V --> V = V0/κ --> V < V0 for κ > 1 i.e. the voltage across the plates decreases when the dielectric is inserted if charge is held constant

19.5 Capacitors and Dielectrics

ENERGY STORAGE IN A CAPACITOR

( )221Energy Ed

dAo !"

#$%

&=κε

221

VolumeEnergydensity Energy Eoκε==

Ad= Volume

Capacitors store charge at some potential difference, therefore they store EPE, since by definition for a point charge q, EPE = qV . Your book shows that the energy stored in a capacitor can be written several ways:

Energy = 1/2 qV = 1/2 CV2 = q2/(2C) --> useful forms to solve problems

Using Energy = 1/2 CV2, C = κε0A/d, and V = Ed , the energy of a capacitor can be written as,

We can then calculate the energy density of a capacitor as,

(valid in general for the energy density stored in an electric field)

Example. How much energy is stored in a 10 F capacitor with a potential difference of 3 V across it?

Energy = 1/2 CV2 = 1/2(10)(3)2 = 45 J This is the same amount of energy stored in a 4.6 kg mass suspended 1 m above the ground.

Chapter 20

Electric Circuits

20.1 Electromotive Force and Current

In an electric circuit, an energy source and an energy consuming device are connected by conducting wires through which electric charges move.

20.1 Electromotive Force and Current

Within a battery, a chemical reaction occurs that transfers electrons from one terminal to another terminal. The maximum potential difference across the terminals is called the electromotive force (emf). The emf has units of Volts, e.g. a 12 V battery has an emf of 12 V

20.1 Electromotive Force and Current

The electric current is the amount of charge per unit time that passes through a surface that is perpendicular to the motion of the charges.

tqIΔ

Δ=

One coulomb per second equals one ampere (A).

20.1 Electromotive Force and Current

If the charges move around the circuit in the same direction at all times, the current is said to be direct current (dc) --> e.g. simple circuits with batteries are normally dc. If the charges move first one way and then the opposite way, the current is said to be alternating current (ac) --> e.g. the current coming out of your electric outlet is ac.

20.1 Electromotive Force and Current

Example 1 A Pocket Calculator The current in a 3.0 V battery of a pocket calculator is 0.17 mA. In one hour of operation, (a) how much charge flows in the circuit and (b) how much energy does the battery deliver to the calculator circuit?

(a)

(b)

( ) ( )( ) C 61.0s 3600A1017.0 3 =×=Δ=Δ −tIq

( )( ) J 8.1V 0.3C 61.0ChargeEnergy Charge Energy ==×=

ΔV = -W/q

20.1 Electromotive Force and Current

Conventional current is the hypothetical flow of positive charges that would have the same effect in the circuit as the movement of negative charges that actually does occur.

20.2 Ohm’s Law

The resistance (R) is defined as the ratio of the voltage V applied across a piece of material to the current I through the material.

It is found experimentally that the current flowing through a circuit is proportional to the voltage across the circuit, i.e., I is proportional to V

20.2 Ohm’s Law

OHM’S LAW The ratio V/I is a constant, where V is the voltage applied across a piece of material and I is the current through the material:

SI Unit of Resistance: volt/ampere (V/A) = ohm (Ω)

IRVRIV

=== or constant

Resistance

20.2 Ohm’s Law

To the extent that a wire or an electrical device offers resistance to electrical flow, it is called a resistor.

Resistance, R, represents to what extent the current can flow freely in the circuit, i.e. the larger R, the more the electrons scatter with atoms in the material. These scatterings slow down electrons and transfer energy as heat to the material.

20.2 Ohm’s Law

Example 2 A Flashlight The filament in a light bulb is a resistor in the form of a thin piece of wire. The wire becomes hot enough to emit light because of the current in it. The flashlight uses two 1.5-V batteries to provide a current of 0.40 A in the filament. Determine the resistance of the glowing filament.

Ω=== 5.7A 0.40

V 0.3IVR