A few important applications of electrostatics are given below :

(i) Electrostatic generators can produce voltages as high as 106 volts. Such high voltages are required for X-ray work and nuclear bombardment.

(ii) We use principles of electrostatics for spray of paints, powder, etc.

(iii) The principles of electrostatics are used to prevent pollution.

(iv) The problems of preventing sparks and breakdown of insulators in high voltage engineering

are essentially electrostatic.

(v) The development of lightning rod and capacitor are the outcomes of electrostatics.

Electric Lines of Forces - The electric field around a charged body is represented by imaginary lines, called electric lines of force.

Properties of Electric Lines of Force

(i) The electric field lines are directed away from a positive charge and towards a negative charge so that at any point, the tangent to a field line gives the direction of electric field at that point.

(ii) Electric lines of force start from a positive charge and end on a negative charge.

(iii) Electric lines of force leave or enter the charged surface normally.

(iv) Electric lines of force cannot pass through a conductor. This means that electric field inside a conductor is zero.

(v) Electric lines of force can never intersect each other. In case the two electric lines of force intersect each other at a point, then two tangents can be drawn at that point. This would mean two directions of electric field at that point which is impossible.

(vi) Electric lines of force have the tendency to contract in length. This explains attraction between oppositely charged bodies.

(vii) Electric lines of force have the tendency to expand laterally i.e. they tend to separate from each other in the direction perpendicular to their lengths. This explains repulsion between two like charges.

Electric Flux Density (D)

The electric flux density at any section in an electric field is the electric flux crossing normally per unit area of that section

Electric flux density, D =ψ/ A or Q/A

The SI unit of electric flux density is *C/m2

Capacitance

The ability of a capacitor to store charge is known as its capacitance. It has been found experimentally that charge Q stored in a capacitor is directly proportional to the p.d. V across it

Q ∝ V

Q/V = Constant = C (coulomb/volt)

The constant C is called the capacitance of the capacitor.The ratio of charge on capacitor plates to the p.d. across the plates is called capacitance of the capacitor.

Unit of capacitance farad (Symbol F) in honor of Michael Faraday.

Breakdown Voltage or Dielectric Strength

In an insulator or dielectric, the valence electrons are tightly bound so that no free electrons are available for current conduction. However, when voltage applied to a dielectric is gradually increased, a point is reached when these electrons are torn away, a large current (much larger than the usual leakage current) flows through the dielectric and the material loses its insulating properties.Usually, a spark or arc occurs which burns up the material.

• The minimum voltage required to breakdown a dielectric is called breakdown voltage or dielectric strength.

• The maximum voltage which a unit thickness of a dielectric can withstand without being

punctured by a spark discharge is called dielectric strength of the material.

• The dielectric strength (or breakdown voltage) is generally measured in kV/cm or kV/mm.

Unit II(B)- AC Fundamentals

Mathematical and graphical representation sinusoidal quantities

The sinusoids graphically within the spacial or phasor-domain form by using Phasor Diagrams, and this is achieved by the rotating vector method.

Basically a rotating vector, simply called a “Phasor” is a scaled line whose length represents an AC quantity that has both magnitude (“peak amplitude”) and direction (“phase”) which is “frozen” at some point in time.

A phasor is a vector that has an arrow head at one end which signifies partly the maximum value of the vector quantity ( V or I ) and partly the end of the vector that rotates.

Generally, vectors are assumed to pivot at one end around a fixed zero point known as the “point of origin” while the arrowed end representing the quantity, freely rotates in an anti-clockwise direction at an angular velocity, ( ω ) of one full revolution for every cycle. This anti-clockwise rotation of the vector is considered to be a positive rotation. Likewise, a clockwise rotation is considered to be a negative rotation.

Although the both the terms vectors and phasors are used to describe a rotating line that itself has both magnitude and direction, the main difference between the two is that a vectors magnitude is the “peak value” of the sinusoid while a phasors magnitude is the “rms value” of the sinusoid. In both cases the phase angle and direction remains the same.

The phase of an alternating quantity at any instant in time can be represented by a phasor diagram, so phasor diagrams can be thought of as “functions of time”. A complete sine wave can be constructed by a single vector rotating at an angular velocity of ω = 2πƒ, where ƒ is the frequency of the waveform. Then a Phasor is a quantity that has both “Magnitude” and “Direction”.

As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate one complete revolution of 360^o or 2π representing one complete cycle. If the length of its moving tip is transferred at different angular intervals in time to a graph as shown above, a sinusoidal waveform would be drawn starting at the left with zero time. Each position along the horizontal axis indicates the time that has elapsed since zero time, t = 0. When the vector is horizontal the tip of the vector represents the angles at 0^o, 180^o and at 360^o.

Likewise, when the tip of the vector is vertical it represents the positive peak value, ( +Am ) at 90^o or π/2 and the negative peak value, ( -Am ) at 270^o or 3π/2. Then the time axis of the waveform represents the angle either in degrees or radians through which the phasor has moved. So we can say that a phasor represent a scaled voltage or current value of a rotating vector which is “frozen” at some point in time, ( t ) and in our example above, this is at an angle of 30^o.

Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the Alternating Quantity at some particular instant in time especially when we want to compare two different waveforms on the same axis. For example, voltage and current. We have assumed in the waveform above that the waveform starts at time t = 0 with a corresponding phase angle in either degrees or radians.

But if a second waveform starts to the left or to the right of this zero point or we want to represent in phasor notation the relationship between the two waveforms then we will need to take into account this phase difference, Φ of the waveform. That is call phase difference which we will study futher.

Definitions: Waveform, Periodic waveform, Cycle, Period, frequency,

Angular velocity, Instantaneous value, Peak (maximum) value

Waveform-The shape obtained by plotting the instantaneous ordinate values of either voltage or current against time is called an AC Waveform. An AC waveform is constantly changing its polarity every half cycle alternating between a positive maximum value and a negative maximum value respectively with regards to time with a common example of this being the domestic mains voltage supply we use in our homes.

Periodic Waveform- This means then that the AC Waveform is a “time-dependent signal” with the most common type of time-dependant signal being that of the Periodic Waveform. The periodic or AC waveform is the resulting product of a rotating electrical generator.

Time Period (T) -The time taken by the waveform to complete one full cycle is called the Periodic Time of the waveform, and is given the symbol “T”. .

Frequency (ƒ) - The number of complete cycles that are produced within one second (cycles/second) is called the Frequency, symbol ƒ of the alternating waveform. Frequency is measured in Hertz, ( Hz ) named after the German physicist Heinrich Hertz

Amplitude (A) - is the magnitude of the signal waveform measured in volts or amps. Known as its Maximum or Peak value represented by the terms, Vmax for voltage or Imax for current .

Instantaneous value: The value of an alternating quantity (voltage or current) at any instant is called instantaneous value.

Peak value

The maximum value attained by an alternating quantity either voltage or current during one cycle is called the peak value. It is also known as amplitude and denoted as E_max or I_max.

Concept of phase and phase difference: lagging, leading and in phase quantities and phasor representation.

Phase Difference: The phase difference between the two electrical quantities is defined as the angular phase difference between the maximum possible value of the two alternating quantities having the same frequency.

In other words, the two alternating quantities have phase difference when they have the same frequency, but they attain their zero value at the different instant. The angle between zero points of two alternating quantities is called angle of phase differences

As the two waveforms are no longer “in-phase”, they must therefore be “out-of-phase” by an amount determined by phi, Φ and in our example this is 30^o. So we can say that the two waveforms are now 30^o out-of phase. The current waveform can also be said to be “lagging” behind the voltage waveform by the phase angle, Φ. Then in our example above the two waveforms have a Lagging Phase Difference so the expression for both the voltage and current above will be given as below. where, i lags v by angle Φ

The two waveforms can also be said to have a Leading Phase Difference. Voltage leads current by angle of Phi. The phasor diagram is shown below

Above figure shows the waveform and phasor diagram representation.