THE ELECTRIC FIELD

• ELECTRIC CHARGE• COULOMB’S LAW• THE ELECTRIC FIELD

Written by Dr. John K. Dayton

ELECTRIC CHARGE:

Electric charge is a fundamental quantity. The smallest possible charge that can be isolated is given by:

8

191.6 10e C

The unit of electric charge is the coulomb, abbreviated C. There are two types of electric charge. While they may be designated in any manner, it is most convenient to designate them as positive, +, and negative, -. Thus the smallest possible charges are +e and -e. Any electric charge, usually designated as q, is composed of collections of +e or -e. Thus q = (+/-)ne where n is a positive integer. Due to this nature we say electric charge is quantized. This is very important when n is small. When n is large the quantized nature of electric charge is not important and is usually ignored.

Benjamin Franklin

Benjamin Franklin was one of the Founding Fathers of the United States and in many ways was "the First American". A renowned polymath, Franklin was a leading author, printer, political theorist, politician, postmaster, scientist, inventor, civic activist, statesman, and diplomat. As a scientist, he was a major figure in the American Enlightenment and the history of physics for his discoveries and theories regarding electricity. As an inventor, he is known for the lightning rod, bifocals, and the Franklin stove, among other inventions. He facilitated many civic organizations, including Philadelphia's fire department and a university.

January 17, 1706 – April 17, 1790

ELECTRICALLY CHARGE OBJECTS:

There are many occasions when we encounter electrically charge objects. When an object is electrically neutral it contains exactly the same number of electrons and protons. Thus the same number of -e charges and +e charges. Electrons furthest from the nuclei in atoms are held in place most weakly and are often lost to other materials. The objects gaining these lost electrons now have more negative charge than positive charge and we say they are negatively charged. Objects whose atoms lost electrons also have an imbalance of charge due to a deficit of negative charge. We say such objects are positively charged. Protons, which each have +e charge, do not transfer between objects due to the fact they are tightly bound within atomic nuclei. Electrically charged objects that are positively charged have a deficit of electrons, not an excess of protons.

+

-

THE FUNDAMENTAL PRINCIPLE OF ELECTROSTATICS:

Like charges repel each other and

Unlike charges attract each other.

+

+

There is a force of interaction between electric charges that obeys the following rule:

THE PROTON:Electric Charge: +eMass:

THE NEUTRON:Electric Charge: 0Mass:

THE ELECTRON:Electric Charge: -eMass:

v F

+

-

BASIC ATOMIC STRUCTURE:

For our purposes it is convenient to use the Bohr model of the atom. This consists of a dense, massive central core composed of protons and neutrons. About the core are electrons in well defined orbits.

271.67 10 kg

271.67 10 kg

319.11 10 kg

Niels Henrik David Bohr

7 October 1885 – 18 November 1962

Niels Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. Bohr was also a philosopher and a promoter of scientific research.

THE COULOMB INTERACTION, COULOMB’S LAW

1 22

q qF k

r

2

2

98.99 10 NmC

k

q1 and q2 are two interacting electric charges.

r is the distance between the two charges.

k is a constant of proportionality.

F is the magnitude of the force between q1 and q2

To find the direction of F, use the Fundamental Principle of Electrostatics.

14 June 1736 – 23 August 1806

Charles-Augustin de Coulomb

He was best known for developing Coulomb's law, the definition of the electrostatic force of attraction and repulsion, but also did important work on friction. The SI unit of electric charge, the coulomb, was named after him.

His name is one of the 72 names inscribed on the Eiffel Tower.

Example: Charge q1 = 3.0 C is placed at x = 0 cm. Charge q2 = 7.0C is placed at x = 30.0 cm. Where between q1 and q2 should a third charge, q, be placed so that the net force on it is zero?

Click For Answer0cm 30 cm

q1 = 3 0 C q2 = 7.0 Cd x - d

3rd charge (unknown q)

F

Charge q is placed a distance d from q1. Its distance from q2 will be x – d where x is 30cm. The resulting forces on q, from the left and right, will be equal and opposite when d is the correct distance. You must solve for d.

Solution is continued on next slide.

+ +

2

1 222

1 22 2 2

2 2 21 2

2 21 2 1 1

6 2 6 6

2

2

2

2 0

4 10 1.8 10 .27 10 0

4 1.8 .27 0

or - .57 (not a so0. lution)12

C Cm Cm

kqq kqq

d x d

q q

d x xd d

q x xd d q d

d q q q xd q x

d d

d d

d mm

The forces on q from q1 and q2 are equal in magnitude.

Cancel common factors k and q and expand denominator.

Cross multiply.

Render in standard quadratic form.

Simplify and solve using the quadratic formula.

Only d = 0.12m is a valid solution.

CONDUCTORS AND NONCONDUCTORS:

Conductors are materials, including metals, that have large numbers of electrons capable of moving throughout the material virtually as a free electron gas.

PROPERTIES OF CONDUCTORS:•Any net electric charge exists on the outer surface.•Any net electric charge on a conductor will adjust its position until an equilibrium condition is established.•The only mobile charge in a conductor is negative due to mobile electrons.

Nonconductors have none of these properties.

THE ELECTROSCOPE:

The electroscope is a simple device used to prove whether or not an object has excess charge on it. The electroscope has three parts: a jar, a conducting rod, and a pair of thin metal strips. The rod sticks through the top of the jar and holds the two metal strips inside the jar. Usually the rod has a round metal ball on its outer end.

When an object with excess charge is brought near the metal rod, electrons will be moved from or toward the metal strips. This causes the metal strips to acquire excess charge, but of the same type on both strips. In turn this causes the strips to repel and swing away from each other, indicating the presence of excess charge.

2:

THE ELECTRIC FIELD:

Surrounding every electric charge is an electric field. Thus at every point in space surrounding an electric charge there is a vector quantity called the electric field. The electric field is defined as the force per unit coulomb and has SI units N/C. If q1 and q2 are two point charges then the force between them is:

then the magnitude of the electric field of q1 is:

1 22

q qF k

r

12

qE k

r

By definition the electric field at a point in space is: o

FE

q

where is the force on a charge qo placed at that point.F

Electric field vectors point away from positive charges and toward negative charges.

Michael Faraday

22 September 1791 – 25 August 1867Michael Faraday was an English scientist who contributed to the fields of electromagnetism and electrochemistry. His main discoveries include those of electromagnetic induction, diamagnetism and electrolysis. Although Faraday received little formal education, he was one of the most influential scientists in history. It was by his research on the magnetic field around a conductor carrying a direct current that Faraday established the basis for the concept of the electromagnetic field in physics. Faraday also established that magnetism could affect rays of light and that there was an underlying relationship between the two phenomena. He similarly discovered the principle of electromagnetic induction, diamagnetism, and the laws of electrolysis. His inventions of electromagnetic rotary devices formed the foundation of electric motor technology, and it was largely due to his efforts that electricity became practical for use in technology.

The concept of an electric field was introduced by Michael Faraday.

THE SUPERPOSITION PRINCIPLE:

Every point charge has its own electric field.

When two or more point charges are near each other their individual electric fields superimpose producing a net, composite electric field.

If is the electric field of q1 at point P and is the electric field of q2 at point P, then is the net electric field at P given by:

,1PE

,2PE

PE

,1 ,2P P PE E E

The net electric field at a point in space is the vector sum of all individual electric fields produced by point charges.

P

,2PE

,1PE PE

NET iE E

In general:

Note that this is a vector sum.

Electric field lines are used to graphically represent an electric field. The following set of rules apply to electric field lines:

1 Field lines originate on positive charges and terminate on negative charges, otherwise they come in from or go out to infinity.

2 Field lines do not cross other field lines.

3 The number of field lines on a charge is proportional to the charge.

4 The number of field lines in a region of space is proportional to the field strength in that region. The closer together they are - the stronger the field.

5 The electric field at a point in space is tangential to the field line through that point.

6 Electric field lines are perpendicular to all conducting and equipotential surfaces.

7 The electric field inside a conductor in electrostatic equilibrium is zero.

ELECTRIC FIELD LINES:

MORE ABOUT CONDUCTORS:

The following object is a conducting material with a cavity inside. Inside the cavity is a charge Q (how it got there we don’t know or care, it’s just there).

conductor

cavity

charge Q

opposite charge –Q

same charge Q

When Q is placed inside the cavity, electrons in the conductor redistribute so that a charge of –Q lines the inside wall of the cavity and a charge of Q lines the outer surface of the conductor. Thus the net charge inside the cavity will be 0 and the overall net charge will lie on the outer surface.

The red lines represent the electric field. Because the cavity contains charge there is an electric field between these charges. This internal field does not penetrate the conductor. In the conducting material there is no field (no electric field vectors, no field lines). A field extends from the surface due to the surface charge. It is important to note that the net charged inside the cavity is zero when the conductor is in electrostatic equilibrium.

conductor charge placed inside a cavity

electric field between charges inside cavity

region of no field

opposite charge due to mobile electrons

net charge lies on surface

The field of a + charge.

THE ELECTRIC FIELDS OF POINT CHARGES:

Essentially, all electric fields are the result of superposition of point charge electric fields.

q = electric charge producing the field

k = electrostatic constant, 2

2

98.99 10 NmC

k

r = distance field point is from charge

E = magnitude of electric field at the field point

Electric fields point away from positive charges and toward negative charges. This establishes the direction of the electric field.

The electric field outside any spherical charge distribution is the same as that of a point charge with the same total charge.

2

qE k

r

EXAMPLE: q1 = -4.0 C is located at the origin. q2 = +3.0 C is located on the y-axis at 10.0 cm. Calculate the net electric field on the x-axis at 15.0 cm.

+

-

q1 = -4C

10cm

q2 = +3C

15cm

P

r2 = .1803m

r1 = .15m

= 33.69o

2,1,

1

222

69.3315.

1.tan

1803.015.1.

PPP

o

EEE

mmmr

This diagram is the first step in a well planned solution.

Solution continues on next slide.

Click For Answer

2

2

2

2

9 6

61,1 22

1

9 6

52,2 22

2

8.99 10 4 101.5982 10

.15

8.99 10 3 108.2964 10

.1803

NmC N

P C

NmC N

P C

CkqE

r m

CkqE

r m

The direction in standard form for Ep,1 is 180o and for Ep,2 it is 326.31o.

Solution continues on next slide.

6 5,1 ,21.5982 10 180 8.2964 10 326.31o o

p pE at E at

6 6,1

5 5,2

5 5

ˆ ˆ1.5982 10 cos 180 1.5982 10 sin 180

ˆ ˆ8.2964 10 cos 326.31 8.2964 10 sin 326.31

ˆ ˆ9.0790 10 4.6020 10

o op

o op

p

E i j

E i j

E i j

2 25 5 6

51

5

6

9.0790 10 4.6020 10 1.02 10

4.6020 10tan 180

1.02 10 at 2

206.99.07

0

90 10

6.9

NP C

o

oNP C

E

E

Final Answer

The most common error made is ignoring the vector nature of the electric field. This problem can only be solved correctly as a vector problem.

THE ELECTRIC DIPOLE FIELD:

322 2

3

2

,

2

P

p

kqaE

x a

if x a then

kqaE

x

DERIVATION OF THE FIELD ALONG THE PERPENDICULAR BISECTOR OF AN ELECTRIC DIPOLE.

, , 2

2 3

2 2

2 2 sin

22

2

P q P q

p y

p

p

kqE E E

r

E E E

kq a kqaE

r r r

kqaE

x a

32

2 2

sina

r

r x a

12

substitutions used:

Refer to the diagram on the previous slide.

THE UNIFORM ELECTRIC FIELD:

The electric field strength of an infinite, uniform plane of charge, such as on a flat, non-conducting sheet is:

The electric field strength of the charge on an infinite conducting plane is:

o

E

ò

In both cases the electric fields are uniform fields that point perpendicularly outward from positive charge and inward toward negative charge.

= surface charge density ( charge per unit area)

= permittivity of free space = oò2

2

128.85 10 CN m

2 o

E

ò

1

4 o

k

ò

MOTION OF A CHARGED PARTICLE IN AN ELECTRIC FIELD:

A particle with mass m and charge q is shown in a uniform electric field E. If the charge is positive, the force on the charge will be in the direction of the electric field. If it is negative, the force will point in the opposite direction. The charge shown in the diagram must be positive since the force on it is in the direction of the field. The acceleration is always in the direction of the force.

Only in a uniform electric field is the acceleration of a charged particle constant.

F qE

qE ma

qEa

m

EXAMPLE: What is the acceleration of a proton in the vicinity of an infinite plane of charge with surface charge density 2.0 pC/m2?

2

2

2

2

121

12

19 1

27

7

2 101.1

1.08 10

299 102 2 8.85 10

1.6 10 1.1299 10

1.67 10

Cm N

C

C

N

s

N

C

m

mo

E

CF eEa

m

a

m kg

ò

directed away from the plane of charge

Click For Solution

GAUSS’S LAW:

Gauss’s Law is best understood and expressed with calculus but an algebraic version, applicable to very symmetrical charge distributions can be used.

cos enclosed

o

qE A

E ò

The left side is the net perpendicular electric field passing through a closed surface of area A surrounding or enclosing the electric charge qenclosed. It has units N∙m2/C and is known as electric flux, designated E. Electric charge outside of this surface does not contribute to the electric field on the surface.

Johann Carl Friedrich Gauss30 April 1777 – 23 February 1855

Johann Gauss was a German mathematician, who contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy, Matrix theory, and optics.

Gauss was the son of poor working-class parents.

His mother was illiterate and never recorded the date of his birth.

Gauss was a child prodigy. There are many anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his magnum opus, in 1798 at the age of 21, though it was not published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.

Using Gauss’s Law, the following electric fields can be derived:

Outside a point or spherical charge distribution:

Outside a uniform line or cylindrical charge:

A uniform plane of charge:

Inside a uniformly charged non-conducting sphere:

2

kqE

r

2 o

Er

ò

2 o

E

ò

3 o

rE

ò

= linear charge density, = surface charge density, = volume charge density.

The electric field of a uniformly charged, non-conducting sphere of charge with charge density and total charge Q and of radius R, both inside and outside the sphere.

E

r (scale in units of R)R

2

kQE

R

2outside

kQE

r

3insideo

rE

ò

(scale in units of Emax)

3

3 1

4 4 o

Q Qk

Volume R

ò

End of Presentation