ELECTROSTATICS

ELECTROSTATICS

INTRODUCTION

Electrostatics is a branch of physics that deals with study of the electric charges at rest. Electrostatic field is defined as an electric field produced by static electric charges.  The charges are static in the sense of charge amount (it is constant in time) and their positions in space (charges are not moving relatively to each other). Electrostatics, as the name implies, is the study of stationary electric charges. A rod of plastic rubbed with fur or a rod of glass rubbed with silk will attract small pieces of paper and is said to be electrically charged. The charge on plastic rubbed with fur is defined as negative, and the charge on glass rubbed with silk is defined as positive.

Since classical physics, it has been known that some materials such as amber attract lightweight particles after rubbing. The Greek word for amber, ήλεκτρον, or electron, was the source of the word 'electricity'. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulomb's law. Even though electrostatically induced forces seem to be rather weak, some electrostatic forces such as the one between an electron and a proton, that together make up a hydrogen atom, is about 36 orders of magnitude stronger than the gravitational force acting between them.

There are many examples of electrostatic phenomena, from those as simple as the attraction of the plastic wrap to your hand after you remove it from a package to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturing, and photocopier & laser printer operation. Electrostatics involves the buildup of charge on the surface of objects due to contact with other surfaces. Although charge exchange happens whenever any two surfaces contact and separate, the effects of charge exchange are usually only noticed when at least one of the surfaces has a high resistance to electrical flow. This is because the charges that transfer are trapped there for a time long enough for their effects to be observed. These charges then remain on the object until they either bleed off to ground or are quickly neutralized by a discharge: e.g., the familiar phenomenon of a static 'shock' is caused by the neutralization of charge built up in the body from contact with insulated surfaces.

1.      STRENGTH CONSIDERATION IN ELECTROSTATIC FIELD

How is field strength defined at a point in the gravitational field? (As the force per unit mass placed at that point in the field – with units therefore of N kg^-1.) What would therefore be the natural way to extend this definition to the electric field? (As the force per unit charge. Thus it would have units of N C^-1.)

We thus define the electric field strength at a point in a field as:

E = F/Q where E = electric field strength (N C^-1)

F = force on charge Q at that point if the field

Important notes:

• The field strength is a property of the field and not the particular charge that is placed there. For example, at a point where the field strength is 2000 N C^-1, a 1 C charge would fill a force of 2000 N whereas a 1 mC charge would fill a force of 2 N; the same field strength, but different forces due to different charges.
• The field strength is a vector quantity. By convention, it points in the direction that a positive charge placed at that point in the field would fill a force.

As will be explained in the next episode, the unit for electric field strength can also be expressed as volts per metre, V m^-1.

Now, for the non-uniform field due to a point (or spherical) charge, we can use Coulomb’s law to find an expression for the field strength. Consider the force felt by a charge q in the field of another charge Q, where the charges are separated by a distance r:

by Coulomb’s Law.

But E = F/q and so

This is our result for the field strength at a distance r from a (point or spherical) charge Q.

Worked examples: Field strength Potential energy and potential
We now turn to considerations of energy. Again, just as in the gravitational case, we choose to define the zero of electric potential energy at infinity. However, because of the existence of repulsion, we have the possibility of positive potential energy values as well as negative ones.

Consider bringing a positive charge q from infinity towards a fixed, positive charge Q. Because of the repulsion between the charges, we must do work on q to bring it closer to Q. This work is stored as the electric potential energy of q. The same would apply if both charges were negative, due to their mutual repulsion. In both cases therefore, the potential energy of q increases (from zero) as it approaches Q; i.e. electric potential energy of charge q is positive.

If Q and q are of opposite signs, however, they attract each other, and now it would take work to separate them. This work is stored as the electric potential energy of q, and so q’s potential energy increases (toward zero) as their separation increases; i.e. q’s potential energy is negative.

With the aid of integration, we can use Coulomb’s law to find the electrical potential energy of q in the field of Q. The final expression turns out as:

EPE = kQq/r

Where r is the separation of the charges. EPE is measured in joules, J.

2. ELECTROSTATIC FIELD IN VACUUM

As seen before, when a charge is placed into a vacuum in which free charges already exist, it experiences a force from these charges. The forces acting on the charge can be examined by the concept of an electrostatic field. An electrostatic field is a vector field. Consider two free charges in a vacuum at positions A and B, and then introduce a test charge at position C.  The electrostatic field at C is the magnitude and direction of the force per unit charge that acts on the charge at C due to the other fixed free charges in the vacuum.

Figure 1 showing the magnitude and direction of the force acting on the charge at B due to the free charges at A and C. 

If the force acting on the charge at C is f(x), and the charge at C is q, the electrostatic field at C could be defined as:

E(x)=f(x)/q                                                        

The force per unit charge on the test particle at C. The SI units of the field from the definition would therefore be Newtons/Coulomb.

Another concept that is important to review in our discussion of an electrostatic field is coulombs law which describes the forces between two charges in a vacuum. It simply states that the electric field intensity of a positive point charge is in the outward radial direction and has a magnitude proportional to the charge and inversely proportional to the square of the distance from the charge.

Electric Charges

The atom consists of elementary particles: protons, neutrons and electrons. In addition to these particles there are many other elementary particles in the universe. These particles constitute matter in the universe, and arrive on Earth in the form of cosmic rays. Elementary particles are also produced on Earth inside physics laboratories by using powerful accelerators. There are over 400 elementary particles known at present, and this number increases every year as the power of the accelerators increases. The investigation of elementary particles is on the foremost edge of the modern physics.

Any elementary particle has a set of strictly defined properties, which are the same for the given type of particles and cannot be changed without destroying the particle. In electricity the most important property of the particle, is its electric charge.

Charge Conservation Law

One of the main laws on nature is the charge conservation law, which states that the net electric charge in any isolated system remains constant

This law, along with energy conservation law, rules over conversions in matter. All nuclear reactions obey this law. For example, a neutron can decay into proton and electron, so that the net charge remains zero.

                                              

 Electric force

The fundamental characteristic of an electric charge is its ability to exert a force on another charge. Unlike charges are attracted to each other and like charges are repelled from each other.

Coulomb’s law states that the magnitude of an electric force between charges Q and q, separated by distance r, is given by:

F α q₁q₂

        r²

F = Cq₁q₂

          r ²

Thus, C =    1              = 8.99 x 10⁹NM²/C²

        4πε₀              

Where is a fundamental physics constant called permittivity of vacuum.

             

 Coulomb's law defines the unit of charge. According to this law two charges of 1 coulomb each, separated by a distance of 1 meter, experience a force given by

This is a very large value, so the charge 1 coulomb is extremely large. The charges of usual objects, as a rule, are measured in nanocoulombs (), or microcoulombs ().

Electric force is vector defined by its magnitude and direction. Let us direct the distance vector, , from Q to q.

Now the electric force can be expressed in vector form, and the Coulomb's law in vector form becomes

Where the force is produced on q by Q

Here the positive sign of corresponds to the repulsive force directed as shown in the above diagram. The negative sign of corresponds to attraction when the direction of force is opposite to that shown in the diagram.

It is important to note that the above Coulomb's law is valid for point charges, when the dimensions of charges are much less than the distance between them. We will show below that the Coulomb's law is also valid for uniformly charged spheres.

Typical problems related to electric fields:

Problem 1

Find force between a proton and an electron placed at the distance 1 μm. (Given that the charges = 1.6 X 10^-19C)

Solution

The force between two oppositely charged particles is attraction and its value is:

q₁&q₂  = 1.6 X 10^-19C

r = 1Nm = 1 X 10^-6 = 10^-6

F = C q₁q₂  = 8.99×10⁹ X 1.6 X 10^-19 X 1.6 X 10^-19  = 23.0144 X 10^-29   = 2.30 X 10^-16 N

r²                                 (10^-6)²                                          10^-12

Problem 2

A positively charged particle with Q= 5 mC is placed between two negatively charged particles with

 q1= 1 mC (left) and q2=9 mC (right). The distance between q1and q is 5 cm and the distance from q to q2 is 9 cm. What is the total force acting on the middle particle? Find the value and the direction.

Solution

The value of the force from particle 1 on the middle particle is

q₁ = 1mC = 1.0 X 10^-3C

q₂ = 5mC = 5.0 X 10^-3C

r  = 5cm = 5 X 10^-2m

F = C q₁q₂  = 8.99×10⁹ X 1.0 X 10^-3 X 5.0 X 10^-3  =           44.95 X 10³         = 1.0 X 10⁷ N

r²                                 (5.0 X 10^-2)²                            25 X 10^-4

 THE ELECTRIC FIELD

In accordance with Coulomb's law, any charge Q produces a force field around itself, which is called the electric field. If this charge is immovable, the electric field is called electrostatic field. This field can be measured by a small test charge q fixed at any point at distance  from the charge Q. According to Coulomb's law the force on the test charge is directly proportional to its charge, so the ratio of this force to the value of the test charge does not depend upon the test charge q and is the unique characteristics of charge Q. This ratio is called the electric field intensity, , or just electric field, defined as the following vector

Thus the electric field is equal to the electric force per unit charge placed in this field. The unit of the electric field is Newton per coulomb

The other unit of the electric field, frequently used, is volt per meter. We will show further that these units are the same.

Using Coulomb's law we get the vector of the electric field produced by a point charge Q

With magnitude

Now we can see that this field does not depend upon the test charge q and depends only on the charge producing this field and the distance where it is measured.

The vector of this electric field is directed from the charge Q for positive charge and toward the charge for negative charge. This is shown in the diagram below at an arbitrary point P

 

Any electric field can be defined graphically by means of the electric field lines, as shown below

The electric field lines are drawn as curves so that the tangent line to the curve at arbitrary point P is directed along the vector of the electric field at this point, and the density of lines is directly proportional to the magnitude of the electric field

Where N is the number of lines crossing a small area A oriented normally to the electric field with the center at the point P, and s is an insignificant arbitrary scale parameter the same for all points.

Taking s = 1 we can rewrite the above formula in form

Where the sign "" means numerical equality without taking units into account

The electric field with  constant everywhere in both the magnitude and the direction is called a uniform electric field. The electric field lines of uniform field are shown below

According to above formula the uniform electric field has a constant density of the electric field lines.

The electric field from a point charge is not uniform. Here the electric field lines are directed radially as shown below for positive (Q>0) and negative (Q<0) charges respectively

Applying formulas for magnitude of electric field and lines density , we get the density of field lines

Thus the electric field of a point charge has radial symmetry. Using , we get the total number of electric field lines for the electric field of a point charge

We got very important result for the point charge, that the total number of electric field lines is defined only by the value of the charge producing this electric field.

3. ELECTROSTATIC FIELD IN MATERIAL MEDIA

Electrostatic fields in free space, produced exclusively by free charges, either by a specified charge distribution or by a free charge on the surface of conductors, but not inside a material media. In this chapter it will be considered the most common case, where materials do not have free charges (ideal dielectric material), as well as the case of free charges considered on conductor materials. Actually, a dielectric is composed of charged particles (the atomic nucleus and electrons), which are strongly joined and which form atoms or molecules. They just change their positions lightly, with movements on the order of the radius of an atom, or one angstrom, (∼1A˙=10−10 m) as a response to external electric fields. This kind of charge is called bound charge, in contrast to free charge found in conducting materials, to express the fact that these charges are not free to move very far or to be extracted from the dielectric material. Strictly speaking, dielectrics do not satisfy this definition, because they have some conductivity, but very little compared to those of metal conductors (more or less 1020 times lower). It can be said that dielectrics are non-conductor materials, or insulators.

Problems A

The square plate made of a dielectric material shown in Fig. 3.4 has thickness e and is polarized over its entire volume according to equation P= (ay3+b)j, where a and b are constants. (a) Determine the polarization surface charge density and the polarization volume charge density. (b) Verify explicitly why the total polarization charge is null.

Solution

(a) Polarization surface density σp is calculated from (3.3). Since vector P has a component just in the uy-direction, it is only necessary to calculate polarization density on the upper and lower surfaces of the plate. On the other ones, the scalar product of (3.3) is zero, because P and n are perpendicular at all points. In Fig. 3.5 The orientation of P at different points can be observed, and also the outward unit normal on surfaces where polarization is not zero. It can be observed on the upper surface of the plate n=+uy and on the lower one, n=−uy. If (3.3) is applied it results, for the upper surface, where y=L/2,

σp,upp=P⋅n=[a(L2)3+b]uy⋅uy=aL38+b.

And for the lower surface

σp,low=P⋅n=[a(−L2)3+b]uy⋅(−uy)=aL38−b.

Problem 2

Fig. 3.5

Vectors P

and n

in the plate of Problem 3.1

To calculate the polarization volume charge density ρp

, (3.2) is applied:

ρp=−∇⋅P=−(∂Px∂x+∂Py∂y+∂Pz∂z)=−3ay2.

(b) To verify that total polarization is null, we calculate this charge:

qp=∫∂VσpdS+∫VρpdV=∫Supp(aL38+b)dS+∫Slow(aL38−b)dS+∫V−3ay2dV.

Functions inside the surface integrals are constant. To solve the volume integral, since the function to be integrated only depends on y, the differential volume can be taken as shown in Fig. 3.5, dV=Ledy. It results,

qp==(aL38+b)Supp+(aL38−b)Slow+∫L/2−L/2−3ay2Ledy=(aL38+b)Le+(aL38−b)Le−3aLey33∣∣∣L/2−L/2=aL44e−aL48e−aL48e=0.

APPLICATIONS OF ELECTROSTATIC 

Photocopier
            An electrostatic copier works by arranging positive charges in a pattern to be copied on the surface of a non-conducting drum, and then gently sprinkling negatively charged dry toner particles onto the drum. The toner particles temporarily stick to the pattern on the drum and are later transferred to the paper and ‘melted’ to produce the copy.

Spray Painting
            In spray painting, particles of paint are give positive charge as they leave the nozzle of a spray gun. The object to be painted is earthed so that there is an electric field between the nozzle and the object. The charged paint droplets follow the field lines are are deposited evenly over the surface of the object.

 Electrostatic Precipitator
            Tiny particles of soot, ash, and dust are major components of the airborne emissions from fossil fuel-burning power plants and from many industrial processing plants. Electrostatic precipitators can remove nearly all of these particles from the emissions.

The flue gas containing the particles is passed between the series of positively charged metal plates and negatively charged wires. The strong electric field around the wires creates negative ions in the particles. The negatively charged particles are attracted by positively charged plates and collect on them. Periodically, the plates are shaken so that the collected soot, ash, and dust slide down into a collection hopper.

 CONCLUSION

An electrostatic field is an electric field produced by static electric charges. The charges are static in the sense of charge amount (it is constant in time) and their positions in space (charges are not moving relatively to each other). Due to its simple nature, the electrostatic field or its visible manifestation – electrostatic force - has been observed long time ago. Even ancient Greeks knew something about a strange property of amber that attracts (under certain conditions) small and light pieces of matter in its vicinity. Much later this phenomenon has been understood and explained as an effect of the electrostatic field. From this historical viewpoint, it would be logical to start the presentation of electromagnetic field theory with electrostatic field. Another reason, as it will be later clear, is its simplicity but also applicability. Namely, electrostatic field plays an important role in modern design of electromagnetic devices whenever a strong electric field appears.

For example, an electric field is of paramount importance for the design of X-ray devices, lightning protection equipment and high-voltage components of electric power transmission systems, and hence an analysis of electrostatic field is needed. This is not only important for high-power applications. In the area of solid-state electronics, dealing with electrostatics is inevitable. It is sufficient to mention only the most prominent examples, such as resistors, capacitors or bipolar and field-effect transistors. Concerning computer and other electronic equipment, the situation seems to be similar: cathode ray tubes, liquid crystal display, touch pads etc.

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