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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, q_{1} and q_{2}, 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 are q_{1} and q_{2} and they are separated by a distance r (Fig. 1), the electric force between the objects has a magnitude F = k q_{1} q_{2}/r^{2} …….(1) Equation 17.3 is called Coulomb’s law. The constant k has the value k = 8.99 ×10^{9} N.m^{2}/C^{2} …….(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 = q_{1}q_{2}/4π ε_{0}r^{2} …….(3) where ε_{0} is yet another physical constant called the permittivity of free space, having the value ε_{0}=8.85×10^{-12} C^{2}/N.m^{2 } (4) The values of ε_{0} and k are related by 1/4π ε_{0} =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 q_{1}q_{2} in the numerator in Equations 1 and 3. The factor q_{1}q_{2} is positive for like charges, so F is positive and the force tends to push the charges farther apart. For unlike charges the product q_{1}q_{2} 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/r^{2} 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.Q_{1}.Q_{2}/r^{2} Where Q_{1} represents the quantity of charge on object 1 and Q_{2 }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 10^{9} N • m^{2 }/ C^{2}. 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 Q_{1} and Q_{2} 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 q_{1}=q_{2}=q According to Coulomb's law, F=(9×10^{9}) (q_{1}q_{2}/r^{2}) 3.7 × 10^{-9}=9 ×10^{9}× [q^{2}/(5×10^{-10})^{2}] q^{2}=[3.7×10^{-9}×(5×10^{-10})^{2}] / (9×10^{9}) q^{2}=10.28×10^{-38 }C q=3.2×10^{-19}C The charge of the electron is given by 1.6×10^{-19}C 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^{-6}C 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^{-6}C 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. Related Resources:-

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, q_{1} and q_{2}, 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”.

If the charges are like, then the forces are repulsive. If the charges are opposite, then the forces are attractive.

F = k q_{1} q_{2}/r^{2} …….(1)

Equation 17.3 is called Coulomb’s law. The constant k has the value

k = 8.99 ×10^{9} N.m^{2}/C^{2} …….(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 = q_{1}q_{2}/4π ε_{0}r^{2} …….(3)

where ε_{0} is yet another physical constant called the permittivity of free space, having the value

ε_{0}=8.85×10^{-12} C^{2}/N.m^{2 } (4)

The values of ε_{0} and k are related by

1/4π ε_{0} =k

so these two forms of Coulomb’s law, Equations 1 and 3 are completely equivalent.

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 q_{1}q_{2} in the numerator in Equations 1 and 3. The factor q_{1}q_{2} is positive for like charges, so F is positive and the force tends to push the charges farther apart. For unlike charges the product q_{1}q_{2} 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/r^{2} 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.

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.Q_{1}.Q_{2}/r^{2}

Where Q_{1} represents the quantity of charge on object 1 and Q_{2 }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 10^{9} N • m^{2 }/ C^{2}.

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 Q_{1} and Q_{2} 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.

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.

(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 q_{1}=q_{2}=q According to Coulomb's law, F=(9×10^{9}) (q_{1}q_{2}/r^{2})

3.7 × 10^{-9}=9 ×10^{9}× [q^{2}/(5×10^{-10})^{2}] q^{2}=[3.7×10^{-9}×(5×10^{-10})^{2}] / (9×10^{9}) q^{2}=10.28×10^{-38 }C q=3.2×10^{-19}C The charge of the electron is given by 1.6×10^{-19}C 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^{-6}C 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?

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^{-6}C 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?

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.

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