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Applied Physics

Lecture 5Lecture 5 Electrostatics

Electrical energy potential difference and electric potential potential energy of charged

conductors Capacitance and capacitors

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Lightning ReviewLightning Review

Last lecture:

1.1. Flux. Gauss’s law.Flux. Gauss’s law. simplifies computation of electric fieldssimplifies computation of electric fields

2.2. Potential and potential energyPotential and potential energy electrostatic force is conservativeelectrostatic force is conservative potential (a scalar) can be introduced as potential potential (a scalar) can be introduced as potential

energy of electrostatic field per unit chargeenergy of electrostatic field per unit charge

Review Problem: Perhaps you have noticed sudden gushes of rain or hail moments after lightning strokes in thunderstorms. Is there any connection between the gush and the stroke or thunder? Or is this just a coincidence?

cosneto

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PEV V V

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16.2 Electric potential and potential energy 16.2 Electric potential and potential energy due to point chargesdue to point charges

Electric circuits: point of zero potential is defined by Electric circuits: point of zero potential is defined by grounding some point in the circuitgrounding some point in the circuitElectric potential due to a point charge at a point in Electric potential due to a point charge at a point in space: point of zero potential is taken at an infinite space: point of zero potential is taken at an infinite distance from the chargedistance from the chargeWith this choice, a potential can be found asWith this choice, a potential can be found as

Note: the potential depends only on charge of an object, Note: the potential depends only on charge of an object, qq, and a distance from this object to a point in space, , and a distance from this object to a point in space, rr..

e

qV k

r

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Superposition principle for potentialsSuperposition principle for potentials

If more than one point charge is present, their electric If more than one point charge is present, their electric potential can be found by applying potential can be found by applying superposition superposition principleprinciple

The total electric potential at some point P due to several The total electric potential at some point P due to several point charges is the algebraic sum of the electric point charges is the algebraic sum of the electric potentials due to the individual charges.potentials due to the individual charges.

Remember that potentials are scalar quantities!Remember that potentials are scalar quantities!

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Potential energy of a system of point Potential energy of a system of point chargescharges

Consider a system of two particlesConsider a system of two particles

If VIf V11 is the electric potential due to charge q is the electric potential due to charge q11 at a point P, at a point P,

then work required to bring the charge qthen work required to bring the charge q22 from infinity to P from infinity to P

without acceleration is qwithout acceleration is q22VV11. If a distance between P and . If a distance between P and

qq11 is r, then by definition is r, then by definition

Potential energy is Potential energy is positivepositive if charges are of the if charges are of the same same signsign and vice versa. and vice versa.

P A

q1q2

r

1 22 1 e

q qPE q V k

r

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Mini-quiz: potential energy of an ionMini-quiz: potential energy of an ion

Three ions, Na+, Na+, and Cl-, located such, that they form corners of an equilateral triangle of side 2 nm in water. What is the electric potential energy of one of the Na+ ions?

Cl-

Na+ Na+

? Na Cl Na Na Na

e e e Cl Na

q q q q qPE k k k q q

r r r

but : !Cl Naq q

0Nae Na Na

qPE k q q

r

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16.3 Potentials and charged conductors16.3 Potentials and charged conductors

Recall that work is opposite of the change in potential Recall that work is opposite of the change in potential energy,energy,

No work is required to move a charge between two points No work is required to move a charge between two points that are at the same potential. That is, W=0 if Vthat are at the same potential. That is, W=0 if VBB=V=VA A

Recall: Recall: 1.1. all charge of the charged conductor is located on its surfaceall charge of the charged conductor is located on its surface

2.2. electric field, E, is always perpendicular to its surface, i.e. no work is electric field, E, is always perpendicular to its surface, i.e. no work is done if charges are moved along the surfacedone if charges are moved along the surface

Thus: potential is constant everywhere on the surface of a Thus: potential is constant everywhere on the surface of a charged conductor in equilibriumcharged conductor in equilibrium

B AW PE q V V

… but that’s not all!

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Because the electric field is zero inside the conductor, no Because the electric field is zero inside the conductor, no work is required to move charges between any two work is required to move charges between any two points, i.e. points, i.e.

If work is zero, any two points inside the conductor have If work is zero, any two points inside the conductor have the same potential, i.e. potential is constant everywhere the same potential, i.e. potential is constant everywhere inside a conductorinside a conductor

Finally, since one of the points can be arbitrarily close to Finally, since one of the points can be arbitrarily close to the surface of the conductor, the surface of the conductor, the electric potential is the electric potential is constant everywhere inside a conductor and equal to its constant everywhere inside a conductor and equal to its value at the surfacevalue at the surface!!

Note that the potential inside a conductor is Note that the potential inside a conductor is notnot necessarily zero, necessarily zero, even though the interior electric field is always zero!even though the interior electric field is always zero!

0B AW q V V

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The electron voltThe electron volt

A unit of energy commonly used in atomic, nuclear and A unit of energy commonly used in atomic, nuclear and particle physics is electron volt (eV)particle physics is electron volt (eV)

The electron volt is defined as the energy that electron The electron volt is defined as the energy that electron (or proton) gains when accelerating through a potential (or proton) gains when accelerating through a potential difference of 1 Vdifference of 1 V

Relation to SI:Relation to SI:

1 eV = 1.601 eV = 1.601010-19 -19 CC··V = 1.60V = 1.601010-19 -19 J J

Vab=1 V

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Problem-solving strategyProblem-solving strategy

Remember that potential is a scalar quantityRemember that potential is a scalar quantitySuperposition principle is an algebraic sum of potentials due Superposition principle is an algebraic sum of potentials due to a system of chargesto a system of charges

Signs are importantSigns are important

Just in mechanics, only changes in electric potential are Just in mechanics, only changes in electric potential are significant, hence, the point you choose for zero electric significant, hence, the point you choose for zero electric potential is arbitrary.potential is arbitrary.

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Example : ionization energy of the electron Example : ionization energy of the electron in a hydrogen atomin a hydrogen atom

In the Bohr model of a hydrogen atom, the electron, if it is in the ground state, orbits the proton at a distance of r = 5.2910-11 m. Find the ionization energy of the atom, i.e. the energy required to remove the electron from the atom.

Note that the Bohr model, the idea of electrons as tiny balls orbiting the nucleus, is not a very good model of the atom. A better picture is one in which the electron is spread out around the nucleus in a cloud of varying density; however, the Bohr model does give the right answer for the ionization energy

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In the Bohr model of a hydrogen atom, the electron, if it is in the ground state, orbits the proton at a distance of r = 5.29 x 10-11 m. Find the ionization energy, i.e. the energy required to remove the electron from the atom.

Given:

r = 5.292 x 10-11 mme = 9.1110-31 kg

mp = 1.6710-27 kg

|e| = 1.6010-19 C

Find:

E=?

The ionization energy equals to the total energy of the electron-proton system,

E PE KE

22 2182.18 10 J -13.6 eV

2 2e e

e ee

m k ee eE k k

r m r r

The velocity of e can be found by analyzing the force on the electron. This force is the Coulomb force; because the electron travels in a circular orbit, the acceleration will be the centripetal acceleration:

e c cm a F

2 2

,2e e

e vPE k KE m

r with

or2 2

2,e e

v em k

r r or

22 ,e

e

ev k

m r

Thus, total energy is

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16.4 Equipotential surfaces16.4 Equipotential surfaces

They are defined as a surface in space on which the potential is the same for every point (surfaces of constant voltage)

The electric field at every point of an equipotential surface is perpendicular to the surface

convenient to represent by drawing equipotential lines

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a+Q

b-Q

16.6 The definition of capacitance16.6 The definition of capacitance

Capacitor: two conductors (separated by an insulator) usually oppositely charged

The capacitance, C, of a capacitor is defined as a ratio of the magnitude of a charge on either conductor to the magnitude of the potential difference between the conductors Q

CV

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1. A capacitor is basically two parallel conducting plates with insulating material in between. The capacitor doesn’t have to look like metal plates.

Capacitor for use in high-performance audio systems.

2. When a capacitor is connected to an external potential, charges flow onto the plates and create a potential difference between the plates.

+ -

- -

3. Capacitors in circuits

symbols

analysis follow from conservation of energy

conservation of charge

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Units of capacitanceUnits of capacitance

The unit of C is the farad (F), but most capacitors have values of C ranging from picofarads to microfarads (pF to F).

Recall, micro 10-6, nano 10-9, pico 10-12

If the external potential is disconnected, charges remain on the plates, so capacitors are good for storing charge (and energy).

1 1F C V

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A

A

+Q

-Q

d

16.7 The parallel-plate capacitor16.7 The parallel-plate capacitor

The capacitance of a device The capacitance of a device depends on the geometric depends on the geometric arrangement of the conductorsarrangement of the conductors

where where AA is the area of one of is the area of one of the plates, the plates, dd is the separation, is the separation, 00 is a constant called the is a constant called the

permittivity of free spacepermittivity of free space,,

00= 8.85= 8.851010-12 -12 CC22/N·m/N·m22

0

AC

d

0

1

4ek

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A parallel plate capacitor has plates 2.00 m2 in area, separated by a distance of 5.00 mm. A potential difference of 10,000 V is applied across the capacitor. Determine

the capacitancethe charge on each plate

Problem: parallel-plate capacitorProblem: parallel-plate capacitor

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A parallel plate capacitor has plates 2.00 m2 in area, separated by a distance of 5.00 mm. A potential difference of 10,000 V is applied across the capacitor. Determine

the capacitancethe charge on each plate

Given:

V=10,000 VA = 2.00 m2

d = 5.00 mm

Find:

C=?Q=?

Solution:

Since we are dealing with the parallel-plate capacitor, the capacitance can be found as

2

12 2 20 3

9

2.008.85 10

5.00 10

3.54 10 3.54

A mC C N m

d m

F nF

9 53.54 10 10000 3.54 10Q C V F V C

Once the capacitance is known, the charge can be found from the definition of a capacitance via charge and potential difference:

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16.8 Combinations of capacitors16.8 Combinations of capacitors

It is very often that more than one capacitor is used in an It is very often that more than one capacitor is used in an electric circuitelectric circuit

We would have to learn how to compute the equivalent We would have to learn how to compute the equivalent capacitance of certain combinations of capacitorscapacitance of certain combinations of capacitors

C1

C2

C3

C5C1

C2

C3

C4

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1 1 1 2 2 2

1 2 1 2 1 2

1 2

1 21 2

1 2

eq

eq

Q C V Q C V

Q Q Q Q Q QQC

V V V V V V

Q Q QC C C

V V V

+Q1

Q1

C1

V=Vab

a

b

+Q2

Q2

C2

a. Parallel combinationa. Parallel combination

1 2V V V

Connecting a battery to the parallel combination of capacitors is equivalent to introducing the same potential difference for both capacitors,

1 2Q Q Q

A total charge transferred to the system from the battery is the sum of charges of the two capacitors,

By definition,

Thus, Ceq would be

1 2eqC C C

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Parallel combination: notesParallel combination: notes

Analogous formula is true for any number of capacitors,Analogous formula is true for any number of capacitors,

It follows that the equivalent capacitance of a parallel It follows that the equivalent capacitance of a parallel combination of capacitors is greater than any of the combination of capacitors is greater than any of the individual capacitorsindividual capacitors

1 2 3 ...eqC C C C (parallel combination)

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A 3 F capacitor and a 6 F capacitor are connected in parallel across an 18 V battery. Determine the equivalent capacitance and total charge deposited.

Problem: parallel combination of capacitorsProblem: parallel combination of capacitors

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A 3 F capacitor and a 6 F capacitor are connected in parallel across an 18 V battery. Determine the equivalent capacitance and total charge deposited.

+Q1

Q1

C1

V=Vab

a

b

+Q2

Q2

C2Given:

V = 18 VC1= 3 FC2= 6 F

Find:

Ceq=?Q=?

First determine equivalent capacitance of C1 and C2:

12 1 2 9C C C F

Next, determine the charge

6 49 10 18 1.6 10Q C V F V C

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b. Series combinationb. Series combination

1 2V V V

Connecting a battery to the serial combination of capacitors is equivalent to introducing the same charge for both capacitors,

1 2Q Q Q

A voltage induced in the system from the battery is the sum of potential differences across the individual capacitors,

By definition,

Thus, Ceq would be

+Q1

Q1

C1

+Q2

Q2

C2

V=Vab

a

c

b

1 1 1 2 2 2

1 2 1 2 1 2

1 2

1 2

1 21 2

1

1 1 1

eq

eq

Q C V Q C V

V V V V V VV

C Q Q Q Q Q Q

Q Q Q

V V VC C C

1 2

1 1 1

eqC C C

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Series combination: notesSeries combination: notes

Analogous formula is true for any number of capacitors,Analogous formula is true for any number of capacitors,

It follows that the equivalent capacitance of a series It follows that the equivalent capacitance of a series combination of capacitors is always less than any of the combination of capacitors is always less than any of the individual capacitance in the combinationindividual capacitance in the combination

1 2 3

1 1 1 1...

eqC C C C (series combination)

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A 3 F capacitor and a 6 F capacitor are connected in series across an 18 V battery. Determine the equivalent capacitance.

Problem: series combination of capacitorsProblem: series combination of capacitors

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A 3 F capacitor and a 6 F capacitor are connected in series across an 18 V battery. Determine the equivalent capacitance and total charge deposited.

+Q1

Q1

C1

+Q2

Q2

C2

V=Vab

a

c

b

Given:

V = 18 VC1= 3 FC2= 6 F

Find:

Ceq=?Q=?

First determine equivalent capacitance of C1 and C2:

1 2

1 2

2eq

C CC F

C C

Next, determine the charge

6 52 10 18 3.6 10Q C V F V C