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CHAPTER 16 Electrical Potential & Electrical Energy

Units

• Electric Potential Energy and Potential Difference • Relation between Electric Potential and Electric Field • Equipotential Lines • The Electron Volt, a Unit of Energy • Electric Potential Due to Point Charges • Potential Due to Electric Dipole; Dipole Moment • Capacitance • Dielectrics • Storage of Electric Energy • Cathode Ray Tube: TV and Computer Monitors, Oscilloscope

Electrostatic Potential Energy and Potential Difference

The change in electric potential energy, PEa – PEb, is when a charge q moves from some point b to a second point a, as the negative of the work done by the electric force to move the charge from b to a.

Moving a positive charge q, against the electric field requires positive work and increases the electric potential energy. Moving a mass m against the gravitational field requires positive work and increases the gravitational PE.

Electric potential is defined as potential energy per unit charge:

Unit of electric potential: the volt (V). 1 V = I J/C. When the electric force does positive work on a charge, the KE increases and the PE decreases. The difference in PE, PEb – PEa, is equal to the negative of the work, Wba, done by the electric field to move the charge from a to b;

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Example 1: An electron in the picture tube of a television set is accelerated from rest through a potential difference

(a) What is the change in electric PE of the electron? (The charge on an electron is:

b) What is the speed of the electron as a result of this acceleration?

The PE doesn’t depend on the mass, only the charge and voltage. Speed does.

Relation between Electric Potential and Electric Field

Solving for the field,

If the field is not uniform, it can be calculated at multiple points:

Example 2: Two parallel plates are charged to produce a potential difference of 50 V. If the separation between the plates is 0.050 m, calculate the magnitude of the electric field in the space between the plates.

Equipotential Lines

An equipotential is a line or surface over which the potential is constant. Electric field lines are perpendicular to equipotentials. (Equipotential lines- (green dashed lines) between two charged parallel plates are always perpendicular to the electric field (solid red lines). The surface of a conductor is an equipotential.

5000b a abV V V V

191.6 10q e x C 19 16( 1.6 10 )( 5000 ) 1.0 10baPE qV x C V x J

31( 9.1 10 )m x kg

KE PE 21

0 ( )2

b a bamv q V V qV

197

31

2 2( 1.6 10 )(5000 )4.2 10 /

9.1 10

baqV x C Vv x m s

m x kg

/ (50 / 0.050 ) 1000 /baE V d V m V m

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Equipotential lines (green, dashed) are always perpendicular to the electric field lines (solid red), shown here for two equal but oppositely charged particles (an “electric dipole”).

The Electron Volt, a Unit of Energy

One electron volt (eV) is the energy gained by an electron moving through a potential difference of one volt.

The joule is a very large unit for dealing with energies of electrons, atoms, or molecules. For this purpose the unit electron volt (eV) is used.

One electron volt is defined as the energy acquired by a particle carrying a charge whose magnitude equals that on the electron (q = e) as a result of moving through a potential difference of 1 V. The electric potential due to a point charge can be derived using calculus.

These plots show the potential due to (a) positive and (b) negative charge.

Electric potential near (a) a positive and (b) a negative charge at the origin. In the case of the + charge, the electric potential forms a “potential hill” near the charge. Near the – a “potential well”.

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Superposition of the Electrical Potential Like many physical quantities, the electric potential obeys a simple superposition principle. The total electric potential due to two or more charges is equal to the algebraic sum of the potentials due to

each charge separately.

Electric Potential Due to Point Charges

Using potentials instead of fields can make solving problems much easier – potential is a scalar quantity, whereas the field is a vector.

Example 3: Potential due to a positive or a negative charge. Determine the potential at a point 0.50 m from a point charge.

From a 20 C point charge.

Example 4: A charge 94.11 10q x C is placed at the origin, and a second charge equal to 2q

is placed on the x axis at the location x = 1.00 m.

(a) Find the electric potential midway between the two charges. Use superposition to write an expression for V at an arbitrary point x between x = 0 and x = 1.00 m:

Substitute numerical values. Note that the midway point between the charges is x = 0.50 m.

9 2 2 9 9 2 2 9(8.99 10 / )(4.11 10 ) (8.99 10 / )( 2)(4.11 10 )

0.500 1.00 0.500

x N m C x C x N m C x CV

m m m

73.9 / 73.9N m C V

(b) The electric potential vanishes at some point between the charges; that is, for a value of x between 0 and 1.00 m. Find this value of x. Set the expression for V in the step before equal to zero, and simplify by canceling the common factor, kq.

20 C

69 2 2 520 10

(9.0 10 / ) 3.6 100.50

Q x CV k x N m C x V

r m

69 2 2 520 10

(9.0 10 / ) 3.6 100.50

Q x CV k x N m C x V

r m

( 2 )

1.00

kq k qV

x m x

( 2 )0

1.00

kq k qV

x m x

1 2

1.00x m x

11.00 2 (1.00 ) 0.333

3m x x x m m

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Example 5: Work done to bring two positive charges close together. What minimum work must be done by an external force to bring a charge 3.00q C from a

great distance away to a point 0.500 m from a charge 20q C ?

Ra = infinity therefore = 0

Potential Due to Electric Dipole; Dipole Moment

Two equal point charges Q, of opposite sign, separated by a distance L, are called an electric dipole. The potential due to an electric dipole is just the sum of the potentials due to each charge, and can be calculated exactly.

Approximation for potential far from dipole:

Or, defining the dipole moment p = Ql,

Capacitance

A capacitor consists of two conductors that are close but not touching. A capacitor has the ability to store electric charge.

Parallel-plate capacitor connected to battery. (b) is a circuit diagram.

( )b a

b a

kQ kQW q V V q

r r

9 2 56 (8.99 10 )(2.00 10 )

(3.00 10 ) 1.08(0.50 )

x N m x CW x C J

m

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When a capacitor is connected to a battery, the charge on its plates is proportional to the voltage:

The quantity C is called the capacitance. Unit of capacitance: the farad (F) 1 F = 1 C/V

The capacitance does not depend on the voltage; it is a function of the geometry and materials of the capacitor. For a parallel-plate capacitor:

The permittivity of free space

Example 6: Capacitor calculations. (a) Calculate the capacitance of a parallel plate capacitor whose plates are 20 cm x 3.0 cm and are separated by a 1.0 mm air gap.

(b) What is the charge on each plate if a 12-V battery is connected across the two plates?

(c) What is the electric field between the plates?

(d) Estimate the area of the plates needed to achieve a capacitance of 1F, given the same air gap d.

Dielectrics

In most capacitors there is an insulating sheet of material, such as paper or plastic, called a dielectric between the plates and is characterized by a dielectric constant K. Capacitance of a parallel-plate capacitor filled with dielectric:

Dielectric strength is the maximum field a dielectric can experience without breaking down.

12 2 28.85 10 /o x C N m

3 212 2 2

3

6.0 10(8.85 10 / ) 53pF

1.0 10o

A x mC x C N m

d x m

12 10(53 10 )(12 ) 6.4 10Q CV x F V x C

4

3

121.2 10 /

1.0 10

V VE x V m

d x m

38 2

12 2 2

(1 )(1.0 10 )10

(9 10 / )o

Cd F x mA m

x C N m

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The molecules in a dielectric tend to become oriented in a way that reduces the external field.

This means that the electric field within the dielectric is less than it would be in air, allowing more charge to be stored for the same potential.

Storage of Electric Energy

A charged capacitor stores electric energy; the energy stored is equal to the work done to charge the capacitor.

Example 7: Energy stored in a capacitor A camera flash unit stores energy in a capacitor at 200 V. How much electric energy can be stored?

Storage of Electric Energy

The energy density, defined as the energy per unit volume, is the same no matter the origin of the electric field:

The sudden discharge of electric energy can be harmful or fatal. Capacitors can retain their charge indefinitely even when disconnected from a voltage source!

Heart defibrillators use electric discharge to “jump-start” the heart, and can save lives.

150 F

21

2PE CV

6 21(150 10 F)(200 )

2x V 3.0J

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Cathode Ray Tube: TV and Computer Monitors, Oscilloscope

A cathode ray tube contains a wire cathode that, when heated, emits electrons. A voltage source causes the electrons to travel to the anode.

The electrons can be steered using electric or magnetic fields.