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IIT JEE Electrostatics Study Material

Electrostatics is a vital branch of Physics. It is an interesting branch and questions are often asked from it in the JEE. It is important to have a strong grip on the topics of electrostatics in order to remain competitive in the JEE.

Introduction:-

The Greek word for amber is “elektron”; this is the origin of the terms electricity and electron.Electrostatic is a branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges with no acceleration. While it’s hard to see the electric charges that are responsible for electricity, it’s easy to see their effects. They’re all around us: in the sparks and shocks of a cold winter day, the imaging process of a xerographic copier, and the illumination of a flashlight when you turn on its switch. Although we often take electricity

for granted, it clearly underlies many aspects of our modern world. Just imagine what life would be like if there were no electric charges and no electricity. For starters, we’d probably be sitting around campfires at night, trying to think of things to do without television, cell phones, or computer games. But before you remark on just how peaceful such a pre-electronic-age existence would be, let me add one more sobering thought: we wouldn’t exist either. Whether it’s motionless as static charge or moving as electric current, electricity really does make the world go ‘round.

Electricity may be difficult to see, but you can easily observe its effects. How often have you found socks clinging to a shirt as you remove them from a hot dryer or struggled to throw away a piece of plastic packaging that just won’t leave your hand or stay in the trash can? The forces behind these familiar effects are electric in nature and stem from what we commonly call “static electricity.” Static electricity does more than just push things around, however, as you’ve probably noticed while reaching for a doorknob or a friend’s hand on a cold, dry day. In this section, we’ll examine

static electricity and the physics behind its intriguing forces and often painful shocks. When a plastic comb is rubbed with your hairs, it acquires the property of attracting light objects such as paper pieces.

EXPERIMENT (Moving Water without Touching It):-

Unlike gravity, which always pulls objects toward one another, electric forces can be either attractive or repulsive. You can experiment with electric forces using a thin stream of water and an electrically charged comb. First, open a water faucet slightly so that the flow of water forms a thin but continuous strand below the mouth of the faucet. Next, give your rubber or plastic comb an electric charge by passing it rapidly through your hair or rubbing it vigorously against a wool sweater. Finally, hold the comb near the stream of water, just below the faucet, and watch what happens to the stream. Is the electric force that you’re observing attractive or repulsive? Why does this force change the path of the falling water? Rubbing the comb through your hair makes it electrically charged. What

other objects can acquire and hold a charge when you rub them across hair or fabric? Which works better: a metal object or one that’s an insulator? Why?

Charge:-

“Charge” is the technical term used to indicate that an object has been prepared so as to participate in electrical forces.” This is to be distinguished from the common usage, in which the term is used indiscriminately for anything electrical. For example, although we speak colloquially of “charging” a battery, you may easily verify that a battery has no charge in the technical sense, e.g., it does not exert any electrical force on a piece of tape that has been prepared. There are two types of electric charge, called positive and negative. The subatomic particle called a proton has a positive charge, and an electron has a negative charge. Charge comes in quantized units. All protons carry the same amount of charge +e, and all electrons carry a charge -e. We will discuss how charge is measured and the unit of electric charge below.

Like charges repel each other, unlike charges attract. The electric force between two objects is repulsive if the objects carry “like” charge, that is, if both are positively charged or both are negatively charged. The electric force is attractive if the two objects carry “unlike” charge. Here the terms like and unlike refer to the signs of the charges, not their magnitudes. So, the expression “like charges” means that the two charges are both positive or both negative.

The expression “unlike charges” means that one charge is positive and the other is negative. Charge is conserved. The total charge on an object is the sum of all the individual charges (protons and electrons) carried by the object. The total charge can be positive, negative, or zero. Charge can move from place to place, and from one object to another, but the total charge of the universe does not change. View this video for more on electrostatics:-

INSULATORS, CONDUCTORS AND SEMICONDUCTORS:- Substances can be classified in terms of their ability to conduct electric charge.

Conductor:- In conductors, electric charges move freely in response to an electric force. All other materials are called insulators and semiconductors. Insulator:- Glass and rubber are insulators. When such materials are charged by rubbing, only the rubbed area becomes charged, and there is no tendency for the charge to move into other regions of the material. In contrast, materials such as copper, aluminium, and silver are good conductors. When such materials are charged in some small region, the charge readily distributes itself over the entire surface of the material. If you hold a copper rod in your hand and rub the rod with wool or fur, it will not attract a piece of paper. This might suggest that a metal can’t be charged. However, if you hold the copper rod with an insulator and then rub it with wool or fur, the rod remains charged and attracts the paper. In the first case, the electric charges produced by rubbing readily move from the copper through your body and finally to ground. In the second case, the insulating handle prevents the flow of charge to ground.

Semiconductor:- Semiconductors are a third class of materials, and their electrical properties are somewhere between those of insulators and those of conductors. Silicon and germanium are well-known semiconductors that are widely used in the fabrication of a variety of electronic devices. When a rod of plastic is rubbed with fur or a glass rod is rubbed against silk, then it is generally observed that the rods start attracting some pieces of paper and seem to be electrically charged. While the charge on plastic is defined to be negative, that on silk is considered positive. The vast amount of charge in an everyday object is usually hidden, comprising equal amount of two kinds – positive and negative. The imbalance is always small compared to the total amounts of positive charge and negative charge contained in the object.

Some general conceptual question:- Question 1: The gift you are about to unwrap is electrically neutral. You tear off the clingy wrapper and find that it has a

large negative charge. What charge does the gift itself have, if any? Answer: It has a large positive charge equal in amount to the wrapper’s negative charge. Why: Since charge is a conserved physical quantity, the wrapper and gift must remain neutral overall even after you separate them. The wrapper’s negative charge must be balanced by the gift’s positive charge. Question 2 : When you peel a piece of adhesive tape off a glass window, you find that the tape is attracted toward the spot it left behind. How did the tape and glass acquire electric charges? Answer: While the tape and glass were in contact, charge was unevenly distributed between their surfaces. Removing the tape merely made that imbalance more obvious. Why: The tape and glass have different chemical affinities for electrons and become oppositely charged whenever they touch. In fact, the tape’s stickiness itself comes from electrostatic attraction. Question 3 : Although any cloud may contain opposite charges, only the violent updrafts inside thunderheads are able to separate those charges and produce lightning. Why does such separation lead to lightning? Answer: That separation takes work, which appears as electrostatic potential energy in the separated charges. The positively charged regions of the thunderhead acquire huge positive

voltages, and the negatively charged regions acquire huge negative voltages. Why: When opposite charges are nearby, they don’t necessarily have much electrostatic potential energy per charge and the voltages may be small. Separating those charges to great distances dramatically increases their stored energy and produces high voltages. Question 4 : The paper in some printing presses moves through the rollers at half a kilometre per minute. If no care is taken, dangerous amounts of static charge can accumulate on parts of the press. How does the moving paper contribute to that charging process? Answer: Contact between dissimilar materials puts charge on the paper, which then carries that charge with it to isolated parts of the press. Enough charge can accumulate on those parts to be dangerous. Why: Nonconductive paper is an excellent transporter of electric charge. Once the paper picks up a static charge by touching a dissimilar material, it can carry that charge with it as it moves through the press. Not surprisingly, printing presses use various tools to suppress this static charging. Question 5 : The conveyor belts used to move flammable materials often have metal threads woven into their fabric. Why are such conducting belts important for fire safety? Answer: An insulating conveyor belt can separate enormous amounts of charge,

leading to high voltages, sparks, and possibly fire. A conductive belt can’t carry charge with it as it moves, so no charge accumulates. Nnn

Coulomb’s Law:-

In 1785 Charles Coulomb (1736–1806) experimentally established the fundamental law of electric force between two stationary charged particles.

An electric force between two point charges (Coulombs Force) has the following properties: 1. It is directed along a line joining the two particles and is inversely proportional to the square of the separation distance r, between them.

2. It is proportional to the product of the magnitudes of the charges, q1 and q2, of the two particles.

3. It is attractive if the charges are of opposite sign and repulsive if the charges have the same sign. The Coulomb’s Law states that,

“The magnitudes of the electrostatic forces between two objects are equal to the Coulomb constant times the product of their two electric charges divided by the square of the distance separating them”. Nature of Force:- If the charges are like, then the forces are repulsive. If the charges are opposite, then the forces are attractive. Description:-

Consider two charged objects that are so tiny that they can

be modeled as point particles. If the charges carried by the two objects areq1 and q2 and they are separated by a distance r (Fig. 1), the electric force between the objects has a magnitude F = k q1 q2/r2 …….(1) Equation 17.3 is called Coulomb’s law. The constant k has the value k = 8.99 ×109 N.m2/C2 …….(2) The direction of the electric force on each of the charges is along the line that connects the two charges and is illustrated in Figure 1 for the case of two like charges and two unlike charges. As already mentioned, this force is repulsive for like charges, corresponding to a positive value of F in Equation 1, whereas the force is attractive for unlike charges and F in Equation 1 is negative in that case. Strictly speaking, the value of F in Equation 1 applies only for two point charges, but it is a good approximation whenever the sizes of the particles are much smaller than their separation r. The mathematical form of Equation 1 is very similar to Newton’s law of gravitation, with the constant k playing a role analogous to the gravitational constant G. Another way to write Coulomb’s law is F = q1q2/4π ε0r2 …….(3) where ε0 is yet another physical constant called the permittivity of free space, having the value ε0=8.85×10-12 C2/N.m2 (4) The values of ε0 and k are related by

1/4π ε0 =k so these two forms of Coulomb’s law, Equations 1 and 3 are completely equivalent.

Features of Coulomb’s Law:- Coulomb’s law has several important properties. 1. We have already seen that the electric force is repulsive for like charges and attractive for unlike charges (Fig. 1). Mathematically, this property results from the product q1q2 in the numerator in Equations 1 and 3. The factorq1q2 is positive for like charges, so F is positive and the force tends to push the charges farther apart. For unlike charges the product q1q2 is negative and the value of F in Equations 1 and 3 is also negative, and the particles are attracted to each other. 2. We have already noted that the form of Coulomb’s law is very similar to Newton’s universal law of gravitation. Both laws exhibit a 1/r2 dependence on the separation of the two particles. Therefore, a negative charge can move in a circular orbit around a positive charge, just like a planet orbiting the Sun, and that was an early model for the hydrogen atom. There is one very important difference, however: gravity is always an attractive force, whereas the electric force in Coulomb’s law can be either attractive or repulsive. 3. The magnitude of F in Equations 1 and 3 is the magnitude of the force

exerted on each of the particles. That is, a force of magnitude F is exerted on charge q1, and a force of equal magnitude and opposite direction is exerted on q2. We should expect such a pair of forces, based on Newton’s third law, the action–reaction principle. Following observations can be noted in connection with Coulomb’s interaction: (a) Basically it is an experimental law and the law is strictly applicable in case of point charges. (b) Coulombs force between two charges is a mutual interaction. It means the force exerted by one charge on the second is equal and opposite to that exerted by second on the first. (c) Coulombs interaction is not affected by the presence of other charges in the neighborhood. (d) It is applicable only to the charges at rest. While dealing with force between charges in motion, the expression has to be modified. (e) Coulomb’s force is attractive in nature for dissimilar charges while it is repulsive in nature for similar charges. (f) The magnitude of force depends upon the magnitude of charges, separation between them and upon the nature of medium in between. (g) The direction force depends upon the relative orientation of the charges. If the charges are like, then the forces are repulsive. If the charges are opposite, then the forces are attractive.

(h) Coulomb’s force is basically a central force. That is, a force acting along the line joining the two charges. (j) Coulomb’s force is conservative in nature. Thus, work done in moving a charge, under the effect of Coulomb’s force, is independent of the path followed. (k) It is based on action-reaction principle.

The following video will further provide you more information on the law:- There exists some force of interaction between the charged particles but it acts over some distance of separation. Whether we consider the case of a plastic tube attracting paper bits or the repulsion between two same charged balloons, there are always two charges with some distance between them. The strength of interaction depends to a large extent on these three variables.

The quantitative expression that describes the influence of these three variables on electric force is known as the Coulomb’s law. In the form of an equation, the law can be written as

F= k.Q1.Q2/r2

Where Q1 represents the quantity of charge on object 1 and Q2 stands for the quantity of charge on object 2 in coulombs. These two quantities are generally expressed as ‘+’ or ‘-‘ which denote positive and negative charge. While negative charge denotes the presence of an excess number of

electrons, the positive charge stands for a shortage of electrons. In terms of force, the negative sign represents a attractive force, the positive sign stands for a repulsive force. The symbol ’r’ represents the distance of separation between the two objects and ‘k’ is the proportionality constant called as the Coulomb’s law constant. This constant is affected by the medium of immersion of charged objects. In particular, for air the value of this k equals 9.0 x 109 N • m2 / C2.

If the medium of propagation or immersion is water then the constant k can also be reduced by a factor of 80. It is clearly evident form the mathematical expression of the coulomb’s law that when the units of k will be substituted into the equation, the units of charge and distance will get cancelled and ultimately, we will be left with Newton as the unit of force. The equation of the law clearly describes the force acting between the objects when they are assumed to be point charges. Although the charge is evenly distributed all throughout the sphere, the center of the sphere can be assumed to be carrying all of the charge. Mathematically, the net force value will be found to be positive if both Q1 and Q2 are of same charges whether both negative or both positive. On the contrary, if one of the charges is positive and other is negative, then the net charge would be negative.

Superposition principle (Net Force Due To a Number of Charges) :-

It states that all the charges when placed near each other behave independent of each other and the net force on one charge due to all other charges is equal to the vector sum of all forces produced by them on the first in accordance with Coulomb’s law. Therefore, force experienced by a given charge in the field of a number of point charges is the vector sum of all the forces.

Electric Lines of Force:-

An electric line of force is defined as the path, straight or curved, along which a unit positive charge is urged to move when free to do so in an electric field. The direction of motion of unit positive charge gives the direction of line of force. The lines of force are straight if the electric field is due to an isolated charge and are curved if the field is due to two or more charges placed near each other. Thus, a line of force may also be defined as a curve, tangent at any point of which gives the direction of the electric intensity at that point.

Properties of Electric Lines Of Force:- (a) The lines of force are directed away from a positively charged conductor and are directed towards a negatively charged conductor. A line of force starts from a positive charge and ends on a negative charge. In other words, line of force starts from higher potential and ends on a lower potential.

(b) Two lines of force never cross each other. If the two lines were to cross, two tangents could be drawn to the lines of force at the common point. This means that there could be two directions of intensity at a point which is impossible. (c) The number of lines of force per unit area (area being normal to lines) is proportional to magnitude of . Thus more concentration of lines represents stronger electric field. (d) One unit of positive charge gives 4π lines of force in free space. Thus, if the lines of force are crowded at a place, it indicates strong field at that place. In the case of a weak field, the lines of force are far apart. Parallel and equally spaced lines of force indicate uniform field. (e) The lines of force meet the surface of a spherical conductor normally. If it were not so, the electric field will have a component parallel to the surface of the conductor. This would mean a flow of current which is absurd. (f) The lines of force never pass through the conductor. This explains the absence of electric field with in the conductor. Let us discuss some of the conceptual questions and problems based on Coulomb’s Law for IIT JEE. Question 1:- After opening your gift, you try to throw away its negatively charged wrapper. However, the wrapper keeps returning to your hand. What attracts it to your electrically neutral hand?

Answer:- Its negative charge polarizes your hand and is then attracted to your hand’s nearby positive charge. Why:- Although your hand is neutral, its charges rearrange in response to the nearby wrapper’s negative charge. Positive charge in your hand shifts toward the wrapper and attracts it. Question 2:- You have two positively charged balls, each of which is experiencing a force of 1 N away from the other. If you halve the distance separating the balls, what force will each exert on the other? Answer:- 4 N. Why:- According to Coulomb’s law, the force on each charge varies inversely with the square of their separation. By halving that separation, you increase the electrostatic Problem 1:-

The electrostatic repulsive force between two positively charged ions carrying equal charges is given by 3.7×10-9 N. These charges are separated by a distance of 5×10-10 m. Calculate the number of electrons missing from each ion? Solution:-

It is given that, F=3.7×10-9 N, r =5×10-10 m and q1=q2=q According to Coulomb's law, F=(9×109) (q1q2/r2)

3.7 × 10-9=9 ×109× [q2/(5×10-10)2] q2=[3.7×10-9×(5×10-10)2] / (9×109) q2=10.28×10-38 C q=3.2×10-19C The charge of the electron is given by

1.6×10-19C Therefore, number of electrons missing from each ion=Total charge of each electron = (3.2×10-19)/ (1.6×10-19) =2

From the above observation we conclude that, the number of electrons missing from each ion would be 2.

Problem 2:-

A particle ‘A’ having a charge of 2 × 10-

6C and a mass of 100g is fixed at the bottom of a smooth inclined plane of inclination 30°. Where should another particle B, having same charge and mass be placed on the incline so that it may remain in equilibrium? Solution:-

First of all draw the F.B.D. of the masses. For equilibrium ∑F = 0 N = mg cos30°

From the above observation we conclude that, the particle B having same charge and mass be placed on the incline will be at 27 cm from the particle A, so that it may remain in equilibrium. Problem 3:-

Two particles A and B having charges 8 x10-6 C and –2 x10-6C respectively are held fixed with a separation of 20 cm. Where a third charged particle should be placed so that it does not experience a net electric force? Solution:-

As the net electric force on C should be equal to zero, the force due to A and B must be opposite in direction. Hence, the particle should be placed on the line AB. As A and B have charges of opposite

signs, C cannot be between A and B. Also A has larger magnitude of charge than B. Hence, C should be placed closer to B than A. The situation is shown in figure. Suppose BC=x and the charge on C is Q

From the above observation we conclude that, the third particle should be placed at 0.2 m from the particle B, so that it does not experience a net electric force.

Electric Field

Intensity

It has been observed that problems of

electro-static can be dealt with more easily if

we introduce a field concept for

electrostaticinteractions. According to this

concept, the two charges need to be in actual

contact for the interactions between them to

take place. The charges are capable of

directly influencing the other charges placed

at a distance, through the intervening

medium.

The region, around any charge, in which its

influence can be realized is known as the

electric field of that charge.

The strength of an electric field is measured

by the force experienced by a unit positive

charge placed at that point. The direction of

field is given by the direction of motion of a

unit positive charge if it were free to move.

The electric field produced by a charge Q

at the location of a small “test” charge q is

defined as the electric force exerted by Q

on q, divided by the test charge q:

So,

In magnitude E =F/q ------(1)

SI Unit: Newton per coulomb (N/C)

Electric field of a point

charge

When a positive test charge is used, the

electric field always has the same direction

as the electric force on the test charge.

In accordance to Coulomb force the force

between two charges Q and q is,

F = kQq/r2 ------(2)

Substitute equation (2) in equation (1), we

get,

E =F/q

= (kQq/r2)/q

= kQ/r2

So, E = kQ/r2

= (1/4πε0)(Q/r2)…… (3)

For several point charges q1, q2, …qn,at

distances r1, r2,….,rn from Q, the resultant

electric field will be,

E (r)=(1/4πε0) Σ(qi/ri 2)ri

Continuous charge

distribution:

The charge is distributed continuously over

some region, the sum becomes an integral.

So, E (r)=(1/4πε0)?(1/r2) dq

If the charge is spread out along a line, with

charge-per-unit-length λ , then dq=(λ)dl

Similarly for surface charge σ,

dq=( σ)da

and

for volume charge ρ,

dq=( ρ )dv

Therefore electric field for line charge will be,

E (r)=(1/4πε0)? (λ (r')/r2) dl

For surface charge,

E (r)=(1/4πε0)? (σ (r')/r2) da

And for volume charge,

E (r)=(1/4πε0)? (ρ (r')/r2) dv

Refer this simulation for

electric field intensity

I use this animation as a lecture aide to

introduce students to the schematic diagram

and e-fields between a plate. Is shows two

metal plates connected to a pair of batteries.

you can click on/off the e-field and show a

charged particle traveling across the plates.

You can click a button labeled, "schematic" to

show the symbols scientists and engineers

use to illustrate the physical setup. Press,

"Play," in the animation to see the charge

accelerate across the plates.

Electric field of a line

charge:-

Fid the electric field a distance z above the

midpoint of a straight line of length 2L, which

carries a uniform line charge λ. From the

figure, the horizontal components of the two

fields cancel, and the net field of the pair is,

cosθ = z/r

r = (z2+x2)3/2

and

dq = λdx

dET = 2dE cosθ

= (2z/4πε0) (λ dx/r3)

So, dET = (2z/4πε0) (λ dx/(z2 +x2) 3/2)

So,

Substitute x =z tanθ, so, dx= zsec2θ dθ in the

above equation, we get,

=(2 λ /4πε0z) [L/(L2+z2)1/2]

So, ET = (1 /4πε0) [2λL/z(L2+z2)1/2]

Case -1

For points far from the line (z>>L), this result

simplifies:

L2+z2 = z2

So, ET = (1 /4πε0) (2λL/z2)

= (1 /4πε0) (q/z2) (Since, q=2λL)

Thus at large distance, the line looks like a

point charge (, q=2λL). Therefore the field

reduces to point chargeq/4πε0 z2.

Case -2

In the limit L → ∞, we obtain the field of an

infinite straight wire:

ET= (2λL)/(4πε0z)(L(1+(z2/L2))

=(2λL/4πε0zL ) (Since, L→ ∞, so

z2/L2 =0)

=(2λ/4πε0z)

In general,

ET= (1/4πε0) (2λ/s)

Here, s is the distance from the wire.

Refer this video to more

about on Electric Field

Electric lines of force

On account of electric intensity at all points in

an electric field, unit positive charge will be

urged to move in a definite direction when

placed in the electric field. The path

described by the unit positive charge is called

the line of force.

An electric line of force is defined as the path,

straight or curved, along which a unit positive

charge is urged to move when free to do so

in an electric field. The direction of motion of

a unit positive charge gives the direction of

line of force.

The lines of force are straight if the electric

field is due to an isolated charge and are

curved if the field is due to two or more

charges placed near each other.

Thus, a line of force may also be defined as

a curve, tangent at any point of which gives

the direction of the electric intensity at that

point.

Electric intensity, at any point, is due to

the interaction of two electro-static fields; one

due to source charge and other due to test

charge.

A line of force starts from a positive charge

and ends on a negative charge. In other

words, line of force starts from higher potential

and ends on lower potential.

Two lines of force never cross each other. If

the two lines were to cross, two tangents could

be drawn to the lines of force at the common

point. This means that there could be two

directions of intensity at a point which is

impossible.

The lines of force meet the surface of a

spherical conductor normally. If it were not so,

the electric field will have a component parallel

to the surface of the conductor. This would

mean a flow of current which is absurd.

The lines of force never pass through the

conductor. This explains the absence of

electric field with in the conductor.

The lines of force tend to contract

longitudinally or lengthwise. That is, they

possess longitudinal tension. Due to this

property the two unlike charges attract each

other.

The lines of force tend to exert lateral

(sideways) pressure. That is, they repel one

another laterally. This explains the repulsion

between two like charges.