Coulomb's Law in Vector Form Derivation
Since force is a vector quantity. So, it is useful to write Coulomb's law in vector form. For this, let us consider that two point charges `q_1` (at A) and `q_2` (at B) are at position vector `\vec r_1`and `\vec r_2` in a vacuum.
`\vec r_1` is the position vector of `q_1`
`\vec r_2` is the position vector of `q_2`
`\vec r_12` is distance from `q_1` to `q_2`
`\vec r_12 = \vec r_1 - \vec r_2`
`\vec r_21` is distance from `q_2` to `q_1`
`\vec r_21 = \vec r_2 - \vec r_1`
`\vec F_12` is force on charge `q_1` due to charge `q_2`
`\vec F_21` is force on charge `q_2` due to charge `q_1`
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| Coulomb's law in vector form derivation |
According to Coulomb's law, electric force on charge `q_1` due to charge `q_2`
`\vec F_12 = \frac{K q_1 q_2}{r_12^2} \hat{r}_12`
`\vec F_12 = \frac{K q_1 q_2}{r_12^2}\times \frac{\vec{r}_12}{r_12}`
`\vec F_12 = \frac{K q_1 q_2}{r_12^3}\times \vec r_12`
`\vec F_12 = \frac{K q_1 q_2}{r_12^3}\times (\vec r_1 - \vec r_2)` ...Eq. (1)
According to Coulomb's law, electric force on charge `q_2` due to charge `q_1`
`\vec F_21 = \frac{K q_1 q_2}{r_21^2} \hat{r}_21`
`\vec F_21 = \frac{K q_1 q_2}{r_21^2}\times \frac{\vec{r}_21}{r_21}`
`\vec F_21 = \frac{K q_1 q_2}{r_21^3}\times \vec r_21`
`\vec F_21 = \frac{K q_1 q_2}{r_21^3}\times (\vec r_2 - \vec r_1)`
`\vec F_21 = - \frac{K q_1 q_2}{r_21^3}\times (\vec r_1 - \vec r_2)`
`\vec F_21 = - \vec F_12` `{\because r_12 = r_21}`
Thus, the force acting due to two charges are equal in magnitude but opposite in direction. So, we can say Coulomb's law is in accordance with Newton's third law of motion. Force is in the direction of the line joining the charges. Thus, constant electric force is the central force.
Important Facts
`\star` For Coulomb's law, charges should be at rest and should be considered as points. The force acting between moving charges cannot be obtained from Coulomb's law because there is an electric force as well as a magnetic force between such charges.
`\star` Coulomb's law is 360not valid for the force between the charges in the range between `10^{- 15}m` or less because in this case, nuclear force is also present.
`\star` The value of Coulomb's force between two charges is unaffected by the presence of other charges as Coulomb's force is a two-body interaction.
`\star` Coulomb's law follows the inverse square law and Coulomb's force is a conservative force.
Coulomb's Law Questions
Q. Coulomb’s Law is valid for ______
(2) For both point charge and distributed charge
(3) Only distributed charges
(4) Neither distributed nor point charge
Answer: (1) Only point charge
Explanation: Coulomb’s Law is valid for only point charges not for distributed charges.
Q. Which one of the following is similar between electrostatic force and gravitational force?
(1) Force can be attractive or repulsive
(2) The force depends on the medium between the bodies
(3) Both forces are strong forces
(4) Force is inversely proportional to the distance between the bodies
Answer: (4) Force is inversely proportional to the distance between the bodies
Explanation: Gravitational force always be attractive and it is a very weak force whereas electrostatic force may be attractive and repulsive both. Gravitational force does not depend on the medium whereas electrostatic force depends upon the medium. But there is a similarity is that both forces are inversely proportional to the distance between them.
Q. Two 1 Coulomb charges are kept at a 1m distance in air medium. Force of attraction or repulsion between them will be ________
(1) `9 \ times 10^9`N
(2) 1 dyne
(3) 1 N
(4) `3 \times 10^3 N`
Answer: (1) `9 \ times 10^9`N
Explanation: According to Coulomb’s Law
`F = \frac{K q_1q_2}{r^2}`
according to question `q_1 = q_2 = 1 C` and r = 1m , So
`F = \frac{K \times 1 \times 1}{1^2}`
Q. Let B be the midpoint of AC. Two point charges Q are placed at A and C. What should be the value of charge placed at B so that the system remains at equilibrium?
(2) –Q/4
(3) +Q/2
(4) +Q/4
Answer: (2) –Q/4
Q. 1 emu = __________ C
(1) 10
(2) `3 \times 10^9`
(3) `4.8 \times 10^{10}`
(4) 0.1
Answer: (1) 10
Explanation: 1 emu charge = 10-coulomb charge.
Q. If the force between two charges is 9N, what will be the force if the distance between them is doubled and both the charges are increased to √2 times?
(1) 9N
(2) 4.5N
(3) 3N
(4) 3.75N
Answer: (2) 4.5N
Explanation: According to Coulomb’s Law
`F_1 = \frac{K q_1q_2}{r^2}`
according to question `q_1 = q_2 = \sqrt 2 q_1 C` and `r_2 = 2 r_1` , So
`F_2 = \frac{K \sqrt 2 q_1 \sqrt 2 q_2}{(2r)^2}`
`F_2 = \frac {2}{4}\frac{K \sqrt 2 q_1 \sqrt 2 q_2}{r^2}`
`F_2 = \frac {2}{4} \times 9 N`
Q. Two charges `q_1` and `q_2` exert some amount of force on each other. What will happen to the force on `q_1` if another charge q3 is brought close to them?
(1) The force will increase
(2) The force will decrease
(3) The force remains the same
(4) The force may increase or decrease depending on whether `q_3` is positive or negative
Answer: (4) The force may increase or decrease depending on whether `q_3` is positive or negative
Q. Two negative charges are kept at a certain distance in the air medium. What will happen if a dielectric slab is inserted between them?
(1) The slab will get heated
(2) Current will flow through the slab
(3) Two charges will attract each other
(4) The net force between the charges will be reduced
Answer: (4) The net force between the charges will be reduced
`F_{medium} = \frac{F_{air}}{\epsilon_r} `
Q. What is the C.G.S. unit of charge?
(1) Stat Coulomb
(2) Coulomb
(3) emu
(4) Pascal
Answer: (1) Stat Coulomb
Q. The dimension of εo _______
(1) `[M^{-1}L^{-3}T^4A^2]`
(2) `[M^{-1}L^{-3}T^4A^4]`
(3) `[M^{-1}L^{-3}T^2A^2]`
(4) `[M^{1}L^{-3}T^4A^2]`
Answer: (1) `[M^{-1}L^{-3}T^4A^2]`
Related Question
1. What is Coulomb's Law and why is it useful to express it in vector form?
2. How can we derive Coulomb's Law in vector form?
3. What are the position vectors involved in Coulomb's Law in vector form?
4. What is the relationship between the distance vectors r12 and r21?
5. What is the equation for the force on charge q1 due to charge q2 in vector form?
6. How does the magnitude of the force depend on the distance between the charges?
7. What is the equation for the force on charge q2 due to charge q1 in vector form?
8. How are the forces on the two charges related to each other?
9. How does Coulomb's Law demonstrate Newton's third law of motion?
10. What are some important facts to know about Coulomb's Law and its limitations?
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