Electric Charges and Fields: Flux, Gauss Law & Coulomb's Law

Content Writer-SME | Updated On - Oct 19, 2024

Electric charge and field is one of the important topics of class 12th physics. Electric charge is one of the seven major fundamental units which is represented by the letter "Q" in Physics Class 12. It can simply be defined as the amount of energy or electrons which pass from one body to another by various modes such as conduction, induction, and more.

Thus, an electric field is created due to electrically charged particles. In this chapter, the fundamental properties and nature of electric charges and fields important laws governing fields have been discussed in detail.

Table of Content

What is Electric Charge?
Conductors and Insulators
Properties of Electric Charge
Coulomb’s Law
Forces Between Multiple Charges
Electric Field
Properties of Electric Field Lines
Electric Flux
Electric Dipole
Gauss Law
Previous Year Questions
Important Topics for JEE Main
Things to Remember
Sample Questions

What is Electric Charge?

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“Electric Charge can be defined as:

“The property of subatomic particles which causes it to undergo a force when situated in an electric and magnetic field is called an electric charge.”

An American scientist, Benjamin Franklin, classified electric charges into two categories on the basis of sign namely- Positive charges and Negative charges. However, in the case, an object possesses no charge, it is then said to be neutral. But, if it does contain an electric charge, it is said to be electrified or charged.

The types of charges can be further defined by protons, electrons, and neutrons. Thus,

Presence of Protons = Positively charged
Presence of Electrons = Negatively charged
Presence of Neutrons = Zero charge

Apart from this, on the basis of polarity, charges are also of two types – like and opposite charges. Like charges repel each other and unlike charges attract each other. It was experimentally observed that if two glass rods are brought close to each other by rubbing them with woolen or silk cloth, they will repel each other. However, the glass rod and the wool will be attracted to each other.

What is 1 Coloumb Charge?

In physics class 12, coloumb is the standard unit of electric charge. One coulomb of charge is defined as:

“One coulomb is the quantity of charge required to carry 1 ampere current in one second.”

Mathematically, the electric charge formula can be given as:

\(Q= I\times T\)

Where,

Q = Electric charge
I = Electric current
T = Time

Electric charge, thus, has the following parameters:

Definition	Electric Charge causes subatomic particles to experience a force when situated in an electromagnetic field.
Symbol	Q
Formula	\(Q= I\times T\)
SI Unit	Coulomb
Other Units	Faraday, Ampere-Hour

Conductors and Insulators

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Conductors are substances that allow electricity to pass through them easily. While the substances which offer high resistance to the passage of electricity through them are called insulators. Examples of conductors are Metals, Sea Water, Graphite, Earth, the Human Body, etc. On the other hand, most non-metals, such as Glass, Porcelain, Plastic, Nylon, And Wood are examples of insulators.

If we bring a charged body in contact with the earth, all the excess charge on the body disappears into the ground through the conductor. This sharing process of the charges with the earth is called grounding or earthing.

Examples of Conductors and Insulators

Examples of Conductors and Insulators

Methods of Charging

The method of supplying or losing electric charge to an object is known as charging. An uncharged object can be charged by means of three methods:

Charging by friction (triboelectric charging)
Charging by conduction
Charging by induction

Charging by Friction

When two objects rub against each other, charge transfer occurs. In this case, one object loses an electron, while the other gains it.The object losing electrons thus becomes positively charged, while the one gaining it becomes negatively charged. Due to friction, both the objects get charged and the method of charging involved in this case is known as charging by friction.

Charging by Conduction

The process of charging an uncharged object by bringing it close to a charged object is called charging by conduction. The charged conductor possesses an unequal number of protons and electrons; thus when an uncharged conductor is brought close to it, it then discharges electrons in order to stabilise itself.

Charging By Induction

Charging by induction is a charging method in which an object gets charged without coming into contact with another charged object. In this case, the charged object is kept near an uncharged conductive material which shall be put on a neutrally charged material. The charge then flows between the two objects and the uncharged conductive material gets charged with opposite polarity.

Charging By Induction

Charging By Induction

Properties of Electric Charge

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If the charged bodies are in very small size compared to the distances between them, then they can be considered as point charges where all the charge content of the body will be concentrated at one point in space. Coulomb is the unit of electric charge. The properties of Electric Charge are:

1. Additivity of Charges

If a system contains n charges q₁, q₂, q₃..., q_n, then the total charge of the system will be q₁ + q₂ + q₃ + ... + q_n. The sign of charge must be included.

2. Charge is Conserved

While rubbing two bodies, the charge gained by one body is similar to the charge lost by another body. In an isolated system consisting of many charged bodies, charges may get redistributed due to interactions among the bodies, but the total charge of the isolated system is always conserved.

3. Quantisation of Charge

The quantization of charge means the electric charge is always an integral multiple of ‘e’. Thus charge ‘q’ on a body is always expressed as

q = ne

The SI unit of a charge is coulomb and is denoted by the symbol C. A coulomb is defined as the amount of charge flowing through a wire in 1 second if the current is 1 A (ampere). In this system, the value of a unit of charge is:

⇒ e = 1.602192 × 10^–19 C

Also Check: Verify the laws of parallel combination of resistances using a metre bridge experiment

Coulomb’s Law

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Coulomb’s law is used to measure the force between two point charges. It is applicable to the linear size of charged bodies which are much smaller than the distance separating them. The size of the charged bodies is so small that it can be treated as point charges.

Thus, Coulomb’s law states that:

“The force of attraction or repulsion between two charged bodies is directly proportional to the product of their charges, while it is inversely proportional to the square of the distance between them.”

Coulomb's Law

Coulomb's Law

Suppose, two point charges q₁, and q₂ are separated by a distance r in a vacuum, the magnitude of the force (F) between them is given by:

\(F= K\times q_1\times q_2 \times \bigg(\frac{1}{r^2}\bigg)\)

Where K is called proportionality constant whose value is equals to:

k = 9 x 10⁹ Nm²C⁻²

Mathematically, the constant k is equalt to \(\frac{1}{4πε_0}\) where:

ε₀ = 8.854 × 10^–12 C² N^–1m^–2

Forces Between Multiple Charges

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Force acted on any charge due to a number of other charges are the vector sum of all the forces on that charge due to the other charges, taken one at a time. The individual forces remain unaffected due to the presence of other charges. This is known as the principle of superposition. According to the superposition principle, "the property of two charges to repel and attract each other remains unaffected despite the presence of a third additional charge".

To calculate the force between multiple charges, we shall use the principle of superposition.

First, evaluate the charges on q₁ due to q₂, q₃, and q₄.

Thus, F₁₂ = ( k q₁q₂ )/ r²₁₂

Herein,

q₁ and q₂ = Charges of q₁ and q₂ respectively
r₁₂ = Distance between q₁and q₂

In the same way,

⇒ F₁₃ = ( k q₁q₃ )/(r₁₃)² ,

⇒ F₁₄= ( k q₁q₄ )/( r₁₄₎²

Thus, it can be concluded from superposition principle, that the total force F on q₁ due to q₂, q₃, and q₄ is

⇒ F = F₁₂ + F₁₃ + F₁₄

Herein,

F₁₂ = ( k q₁q₂ )/ (r₁₂)²,
F₁₃ = ( k q₁q₃)/( r₁₃)²,
F₁₄= ( k q₁q₄ )/( r₁₄)²

Typically, for a system of n particles, possessing q₁, q₂, q₃, q₄, … qn charged particles, the force F on q₁ due to every other particle in the system is:

⇒ F = F₁₂ + F₁₃ + F₁₄+ …. F_1n

Thus, F₁₂, F₁₃, F₁₄… F_1n are the obtained forces on particle q₁ due to q₂, q₃, q₄… qn respectively.

Superposition of Electric Forces Formula

The formula for the superposition of electric forces is:

\(\displaystyle \vec{F}(r)=\frac{1}{4πε_0}Q \sum_{i=1}^N\frac{q_i}{r^2_i}\hat{r_i} \)

Electric Field

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Let a point charge Q be placed in a vacuum, at the origin O and another point charge q at a point P, where OP = r.

Thus, according to Coulomb’s law, the charge Q will experience a force on q.

The electric field due to a charge Q at a point in space is the force that a unit positive charge would experience if placed at that point.
The charge Q produces an electric field in the surrounding.
The electric field produced by the charge Q at a point r is given as

\(E=\frac {1}{4\pi \epsilon_o} \frac {Q}{r^2}\)

The SI unit of the electric field is N/C (Newton/ Coulomb)

There are two types of electric field

Uniform Electric Field

An electric field that has the same strength and direction at every point in a region is called a uniform electric field.

Non-Uniform Electric Field

An electric field that has different strengths and directions at different points in a region is called a non-uniform electric field.

Electric Charges and Fields

Electric Charges and Fields

Properties of Electric Field Lines

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Some of the important properties of Electric Field Lines are:

Field lines form a continuous curve without any breakage in a charge-free region.
Two-electric lines of force never cross each other.
Electric field lines start from the positive charge and end in the negative charge.
Electrostatic field lines never form any closed loops.

Physical Significance of Electric Field

The true physical significance of the concept of the electric field is to deal with time-dependent electromagnetic phenomena.

Thus, the effect of any motion of q₁ on q₂ in an electric field due to a system of charges is the vector sum of the electric fields at the point due to individual charges.
The accelerated motion of charge produces electromagnetic waves, which propagate with the speed c, applying force on other charges.

Electric Flux

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The total number of electric field lines passing a given area in a unit of time is known as the electric flux.

In electromagnetism, electric flux can be defined as the measure of the electric field via a given surface, even though an electric field cannot flow in itself. From the definition, the Electric flux formula can be given by: \(\phi\) = EAcos\(\theta\)
This is proportional to the number of field lines cutting the area element.
Here angle θ here is the angle between E and S.
For a closed surface, θ is the angle between E and the outward normal to the area element.
To calculate the total flux through any given surface divide the surface into small area elements, calculate the flux of each element, and add them up.
Now, the total flux θ through a surface S is defined as θ ~ Σ E. S. The approximation symbol is used because electric field E is constant over the small area element.

Electric Dipole

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A pair of equal and opposite charges separated by a specific distance forms an Electric Dipole.

Electric Dipole Moment

The electric dipole moment is defined as the product of the magnitude of either charge and the distance between the charges. Consider an electric dipole having charges +q and -q separated by distance 2l, then the electric dipole moment of the electric dipole is given as

\(p=q\times 2l\)

It is a vector quantity and its direction is from negative charge to positive charge.

Physical Significance of Dipoles

In CO₂ and CH₄ types of molecules, the dipole moment is zero but a dipole moment is developed when an electric field is applied. But in some molecules, where the centers of negative charges and of positive charges do not coincide have a permanent electric dipole moment. Those show permanent dipole moments even in the absence of an electric field called polar molecules. An example of a polar molecule is a Water molecule, H₂O.

Dipole in a Uniform External Field

There is a force qE and –qE applied on q and –q respectively. The net dipole is zero, so E is uniform. If the charges are separated, and the forces act at different points, a torque is generated on the dipole. The torque (couple) is independent of the origin when the net force is zero.

The magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces).

Thus,

Magnitude of Torque = pE sin θ

Its direction is normal to the plane of the paper, coming out of it. The magnitude of p × E is also pE sinθ and its direction is normal to the paper, coming out of it.

Thus, torque = p × E

Continuous Charge Distribution

The field generated by continuous charge distribution can be obtained in the same way as for a system of discrete charges. Simply, the region where the charges are closely spaced is known to contain Continuous distribution of charge. It can be further divided into three parts:

Linear Charge Distribution
Surface Charge Distribution
Volume Charge Distribution

Gauss Law

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Gauss Law states that the total electric flux through a closed surface is zero if there is no charge enclosed by the surface. Gauss law also states that Electric flux through a closed surface is, \(S=\frac{q}{ε_0}\)

Where q = total charge enclosed by S.

Applications of Gauss Law

Some of the important applications of Gauss law are:

(i) Electric due to thin infinitely long uniformly charged straight wire:

\(E= \frac{λ}{2πε_or}\)

where,

λ = Linear charge density
r = Perpendicular distance of the point from the wire
ε_o = Absolute permittivity

(ii) Electric due to uniformly charged infinite plane sheet:

\(E=\frac{σ}{ 2ε_0}\)

where,

σ = Surface charge density
ε_o = Absolute permittivity

(iii) Electric due to uniformly charged thin spherical shell of radius R:

Outside the shell at distance r, E = 1/4πε_o q/r²

On the surface of the shell, E = 1/4πε_o q/R² = σ/ε₀

Inside the shell, E = 0

Gauss Law Explanation

Gauss Law Video

Previous Year Questions

Important Topics for JEE Main

As per JEE Main 2024 Session 1, important topics included in the chapter Electric Charges and Fields are as follows:

Topics	Number of Questions Asked
Electrical Charge	1
Conductors	1
Electric Field	1

Things to Remember

Electric Charge can be expressed as the amount of energy or electrons passing from one body to another by various modes like conduction, induction, and more.
Electric charges can be further classified into two terms: Positive charges and Negative charges.
Gauss’s Law claims that the total electric flux passing via a closed surface is zero if there is no charge enclosed by the surface.
The SI unit of an electric field can be expressed as N/C (Newton/ Coulomb).

Sample Questions

Ques. Write the orientation of an electric dipole in a uniform electric field that will correspond to stable equilibrium? (1 Mark)

Ans. When dipole moment vector is parallel to electric field vector: PE

Ques. Two point charges having equal charges separated by 1 m distance experience a force of 8 N. What will be the force experienced by them, if they are held in water, at the same distance? (Given : K_water = 80) (2 Marks)

Ans. Force acting between two point charges:

F _air= \(\frac{1}{4 \pi \varepsilon _0} \frac{q_1q_2}{r^2}\)

F _water= \(\frac{1}{4 \pi \varepsilon _0K} \frac{q_1q_2}{r^2}\)

Therefore = \(\frac{F_{air}}{F _{water}} = K\)

\(\frac{8}{F_{water}} = 80\)

F _water= \(\frac{8}{80} = \frac{1}{10} N\)

Ques. The sum of two point charges is 7 micro C. They are further seen to repel one other with a force of 1 N when kept 30 cm apart in free space. Determine the value of each charge. (CBSE Foreign 2009) (2 Marks)

Ans: Given: q1 + q2 = 7 microC

F=1 N

From coulomb’s law, F = \(\frac{1}{4\Pi \varepsilon_0 } . \frac{q1q2}{r2}\)

r=10 cm = 0.1 m , r²=10^-2 m²

So, 1 = 9x10⁹ x 10² x q1q2

q1q2 = 1.1 x 10^-11

q1 + q2 = 7 x 10^-6

Thus, from the above 2 relations, q1 = 2 μC and q2 = 5 μC

Ques. A positive point charge (+q) is kept in the vicinity of an uncharged conduction plate. Sketch electric field lines originating from the point on to the surface of the plate. (CBSE, 2009)(2 Marks)

Ans.

Ques. A metallic sphere is placed in a uniform electric field as shown in the figure. Which path is followed by electric field lines and why? (CBSE, 2010)(2 Marks)

Ans. The path followed is:

Path d is followed by electric field lines

Electric field intensity inside the metallic sphere will always be zero, as

1. No electric lines of force exists inside the sphere,

2. Field will always be normal to the surface.

Ques. A charge q is placed at the center of a cube of side l. What is the electric flux passing through each face of the cube? (2007F, 2010, 2010 F)(2 Marks)

Ans. Total electric flux linked with cube

Φ = \(\frac{q}e\)

As charge is at center, therefore electric flux is symmetrically distributed through all 6 faces.

Flux through each face, \(\frac{1}6\) Φ

= \(\frac{q}{6E_o}\)

Ques. Is Electric Charge a Vector Quantity? (3 Marks)

Ans. Electric charge is considered as a scalar quantity. Alongside possessing a magnitude and direction, a quantity that can be a vector should also follow the laws of vector addition, like the triangle law of vector addition and the parallelogram law of vector addition. Thus, if applicable, then the quantity can be a vector quantity. When two currents are seen to meet at a junction (for an electric current), the resultant current of them will be an algebraic sum rather than a vector sum. Thus, an electric current is a scalar quantity, although it possesses magnitude and direction.

Ques. What are the three Methods of Charging? (3 Marks)

Ans. The three methods of charging are: