Teruo Matsushita - Electricity and Magnetism

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Part I
Static Electric Phenomena

Springer Japan 2014

Teruo Matsushita Electricity and Magnetism Undergraduate Lecture Notes in Physics 10.1007/978-4-431-54526-2_1

1. Electrostatic Field

Teruo Matsushita 1

(1)

Department of Computer Science & Electronics, Kyushu Institute of Technology, Iizuka, Fukuoka, Japan

1.1 Electric Charge in Vacuum

When we touch a metal doorknob after walking on a carpet on a dry day, we sometimes feel a shock on the fingertips as a small crackle. If it is dark, we can see a spark when inserting a key into a keyhole. This is the same phenomenon as thunder. This phenomenon is brought about by electric charge in substances. The usual frictional electricity we experience also comes from electric charges.

Electric charge build-up in a substance that causes various kinds of electric phenomena, including the above examples. Matter is a substance that obeys universal gravity laws, and its magnitude is quantitatively described in terms of mass. In the case of electric phenomena, an amount of electric charge quantitatively describes the phenomena and the same term, electric charge, is also used to mean the amount of electric charge.

Unlike mass, there are two kinds of electric charge, positive and negative. The components of the electric charge are the proton with positive charge and electron with negative charge. The electric charge of a proton is called the elementary electric charge and its magnitude is

where the unit [C] is coulomb . The electric charge of an electron is e . The elementary electric charge is the minimum amount of electric charge, and any electric charge is its integral multiple. Since e is sufficiently small, electric charge can be regarded as a continuous quantity in many cases. This is similar to the fact that an amount of water can be regarded as a continuous quantity in usual cases.

On an atomic scale the nucleus of an atom is composed of protons and neutrons, which are electrically neutral, and electrons stay in orbits around the nucleus. There are innumerable positive and negative electric charges in substances. Since the size of each atom is very small, electrons and protons can be regarded as being in the center of each atom on a macroscopic scale. As a result, the positive and negative charges cancel each other to yield an electrically neutral state. Ionic crystals composed of equal amounts of positive and negative ions can also be regarded as electrically neutral on the macroscopic scale, since the distance between these ions is sufficiently small. Sometimes the electric charge is not balanced. In such a case the electric charge that remains after cancellation causes various electric phenomena.

There are two kinds of electric charge that cause electric phenomena: one is true electric charge , which can be transferred outside a substance and the other is polarization charge , which is locally bound around a nucleus and cannot be transferred outside. The former charge appears on the surface of a conductor and will be covered in . These charges that contribute to electric phenomena are called free electric charge .

Electric charge is generally distributed with some density in the interior or on the surface of matter. Electric charge small enough to be regarded as a point is called point charge . This is similar to a material particle in mechanics. Electric charge distributed along a thin line with negligible cross-sectional area is line charge , and electric charge distributed on a surface with negligible thickness is surface charge .

The principle of conservation of charge is a fundamental principle for electric charge, which is similar to the law of conservation of mass in mechanics. It states that the amount of electric charge is constant in a closed system. Even when positive and negative electric charges cancel each other, resulting in an electrically neutral state, the algebraic sum of electric charge is unchanged.

1.2 Coulombs Law

Electric force works between electric charges, and this force is called the Coulomb force . This force is analogous to universal gravitation between two particles with masses. The Coulomb force on two point charges in vacuum is expressed as follows:

• The force between two electric charges of the same kind (i.e. both positive or both negative) is repulsive and the force between electric charges of different kinds (i.e. one positive and one negative) is attractive.
• The magnitude of the force is proportional to the product of the two electric charges.
• The magnitude of the force is inversely proportional to the square of the distance between the two electric charges.
• The direction of the force lies on the straight line connecting the two electric charges.

The first property is different from the property of universal gravitation whereby the force between two masses is always attractive. The Coulomb force between two point charges, q and q , separated by distance d is mathematically expressed as

(1.1)

where 0 is a constant called the permittivity of vacuum ,

(1.2)

with m/s denoting the speed of light in vacuum. The force in Eq.() is repulsive when F > 0 and attractive when F < 0. This equation is called Coulombs law .

Since force is a vector, the Coulomb force can be expressed as a vector. We denote the direction vector of point charge q measured from the position of q as , as shown in Fig.. Then, its magnitude is and the unit vector pointing from q to q is . Hence, the force that works on q is

(1.3)

The force on q is given by , and the law of action and reaction is satisfied.

Fig. 1.1

The Coulomb force exerted on point charge q by q

When there are more than two material particles, the gravitational force on one particle is the linear sum of the gravitational forces exerted on it by all other particles, and the principle of superposition holds. The same principle holds also for the Coulomb force. Assume that n point charges,

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