Electric Charge, Coulomb's law and Electric Field Test - 1

The correct answer is option 4) i.e. All of the above

CONCEPT:

• Electric charge is a fundamental property of matter that causes the matter to experience a force when kept in electric and magnetic fields.
  • There are two types of charges possessed by matter - negative charge and positive charge.
  • The matter that is neither positive nor negative is said to be neutral.
  • The electric charge of the matter comes from the charge carried by electrons and protons at the atomic level.
  • The SI unit of electric charge is coulomb (C).
• Basic properties of electric charges:
  • ​Additivity of electric charge: The total charge of a system is obtained from the algebraic sum of the constituent charges.
  • Conservation of electric charge: Electric charges obey the law of conservation of charges. According to the principle of conservation of charges, the charges are neither created nor destroyed; they are only transferred from one body to the other. 

  • Quantization of electric charge: According to the principle of quantization of electric charge, all the charges are integral multiples of the charge of a fundamental quantity. An electron or a proton is a fundamentally charged quantity.

    • The charge possessed by a system, Q = ne

​Where n is the number of fundamental quantities and e is the charge of the fundamental quantity.

EXPLANATION:

• All the given statements are the basic properties of electric charges. Hence, they all are true.

CONCEPT:   

Coulomb's law:

• Coulomb's law is a quantitative statement about the force between two-point charges at rest.
• When the linear size of charged bodies is much smaller than the distance separating them, the size may be ignored and the charged bodies are treated as point charges.
• Coulomb measured the force between two point charges and found that it varied inversely as the square of the distance between the charges and was directly proportional to the product of the magnitude of two charges and acted along the line joining the two charges.
• Mathematically it is written as,

\(\Rightarrow F = K\frac{q_1q_2}{r^2}\)

Where q1 = charge on 1st body, q2 = charge on a second body, and r = distance between two charged bodies.

EXPLAINATION:

Coulomb's law is valid when,

• From the above, it is clear that the charge is at rest
• The separation between charge must be greater than nuclear size(10^-15m) because for a distance less than (10-15m), strong nuclear forces dominate over electromotive force.
• From the above, the charges should be Point charge

CONCEPT:

Electric field lines:

• An electric field line is an imaginary line along which a positive test charge will move if left free.
• Electric field lines are drawn to represent the electric field.

Properties of electric field lines:

• Electric field lines start from positive charges and end at negative charges. If there is a single positive charge then electric field lines start from positive charge and end at infinity. Similarly, if there is a single negative charge then electric field lines start from infinity and end at a negative charge.
• In a charge-free region, electric field lines can be taken to be continuous curves without any breaks.

• The tangent at any point on the electric field line gives the direction of the electric field at that point.
• Electric field lines due to a point charge never intersect each other.
• The electric field line never forms a closed loop.
• The density of the electric field lines at a point indicates the strength of the electric field at that point.

EXPLANATION:

• The tangent at any point on the electric field line gives the direction of the electric field at that point.
• Electric field lines due to a point charge never intersect each other.
• If two electric field lines due to a point charge intersect with each other, then two tangents can be drawn at that point in two different directions which shows two different direction of the electric field at that point that is not possible, because at one point in the space the electric field will have only one direction. Hence, option 1 is correct.

CONCEPT:

Superposition Principle: 

• The principle of superposition states that when a number of charges are interacting, the total force on a given charge is the vector sum of the forces exerted on it due to all other charges.
• The force between two charges is not affected by the presence of other charges.

• This is used to calculate the following parameter at the observation point, to any configuration of charges.
  • Net force.
  • Net electric field.

CALCULATION:

If system is in equilibrium, then total force on all charges will be zero.

F₁ + F₂ + F₃ = 0

∴ F₁ = -(F₂ + F₃₎

Force on 10μC charge will be -

CONCEPT:

• Electric charge: It is an intrinsic property of the elementary particles of matter which gives rise to electric force between various objects.
• It is a scalar quantity.
• SI unit of electric charge is coulomb (C).
• The total charge on the body is always an integral multiple of a basic quantum of charge i.e. q = ne

Where n = integer

EXPLANATION:

If a system has n charges like \({q_1},{q_2},{q_3},...{q_n}\).

• Then their total charge will be sum of all charges i.e.

\(\Rightarrow \left( {{q_1} + {q_2} + {q_3} + \ldots {q_n}} \right)\)

• This is called a principle of superposition of charges. Therefore statement 1 correct. 
• It is the property of charges that is, like charges repel each other whereas unlike charges attract each other. Therefore statement 2 is incorrect and statement 3 is correct.

CONCEPT:

Electric dipole in a uniform external field:

• We know that when a charge q is placed in electric field E, it experiences a force F, the force is given as,

⇒ F = qE     -----(1)

• So when an electric dipole is placed in the electric field according to the diagram,

The force on the +q and the -q charge due to the electric field is given as,

⇒ F = qE

• The net force on the electric dipole will be zero.
• The torque on the electric dipole is given as,

⇒ τ = pE.sinθ

\(⇒ \vec{τ} = \vec{p}\times\vec{E}\)

Where θ = angle between the dipole and the electric field

• This torque will tend to align the dipole with the electric field.

EXPLANATION:

• When an electric dipole is placed parallel to the electric field, the angle between the electric field and the dipole will be zero.

⇒ θ = 0

So the torque is given as,

⇒ τ = pEsinθ

⇒ τ = pE sin0°

⇒ τ = 0

• Hence, option 3 is correct.

Electric field due to charge at point P at origin

\({E_1} = \frac{1}{{4\pi {\epsilon_0}}}\frac{Q}{{{r^2}}}\)

\({E_1} = 9 \times {10^9}\frac{{2 \times {{10}^{ - 9}}}}{1} = 18\;V/m\)

Electric field due to charge at point Q at origin

\({E_1} = \frac{1}{{4\pi {\epsilon_0}}}\frac{Q}{{{r^2}}}\)

\({E_2} = 9 \times {10^9}\frac{{ - 2 \times {{10}^{ - 9}}}}{1} = - 18\;V/m\)

Since both charges are opposite to each other, the resultant electric field

E_R = E₁ - E₂ = 18 – (-18) = 36 V/m

Coulomb's law in Electrostatics:

It state’s that force of interaction between two stationary point charges is directly proportional to the product of the charges, and inversely proportional to the square of the distance between them and acts along the straight line joining the two charges.

F ∝ q1 × q2

\(F \propto \;\frac{1}{{{r^2}}}\)

\(F = K\frac{{{q_1}\; \times \;{q_2}}}{{{r^2}}}\)

Where K = constant called electrostatic force constant is given by:

\(K=\frac{1}{4\pi ϵ }\)

\(F = \frac{1}{4\pi ϵ}\frac{{{q_1}\; \times \;{q_2}}}{{{r^2}}}\)

Observation:

• The electrostatic force will increase when 'r' is small, i.e. when the charges are less apart.
• The electrostatic force will increase when the dielectric constant 'ϵ' is small.
• The force also depends on the magnitude of the charges.

CONCEPT:

Electric field intensity: 

• It is defined as the force experienced by a unit positive test charge in the electric field at any point.
• The positive charge experiences a force in the direction of the electric field and the negative charge experiences a force in the direction opposite of the electric field.

\(⇒ E=\frac{F}{q_{o}}\)    

Where E = electric field intensity, qo = charge on the particle

CALCULATION:

Given q_o = -6C, and F = 60 N towards north

• We know that the electric field intensity is given as,

\(⇒ E=\frac{F}{q_{o}}\)

\(⇒ E=\frac{60}{6}\)

⇒ E = 10 N/C

• The negative charge experiences a force in the direction opposite of the electric field.
• So the direction of the electric field is towards the south. Hence, option 2 is correct.

Concept:

The relation between the force and the electric field on a charge q is given by:

F̅  = qE̅ 

E̅ = Electric Field Vector

F̅ = Force vector

q = charge of the particle

Calculation:

Given q = 1 C (unit charge)

F̅ = E̅ 

We conclude that the electric field direction will be the same as the direction of the Coulomb force acting on the unit charge.