Electric Field of a Point Charge: Formula and Derivation

An electric field is a region or space around a charged body within which its influence can be felt by other charged bodies. Mathematically, it can be expressed as electric force per unit charge. 

• The direction of the electric field is radially outward from a positive charge and radially inward to a negative charge.
• Electric charge is a property of subatomic particles that makes them experience a force when placed in a magnetic or electric field.
• A point charge is a hypothetical charge whether positive or negative, whose electric field is found to be at a particular distance. 
• Electric fields arise from time-variable electric currents and electric charges.
• Magnetic and electric fields are both expressions of the electromagnetic field, one of the four basic interactions (also known as forces) of nature.

Key Terms: Coulomb's Law, Electric Field, Electrostatic Field, Electric Charge, Electromagnetic Field, Absolute permittivity

Also Read: Electric Flux

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What is Electric Field?

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Electrical fields (sometimes referred to as E-fields) refer to the physical field that surrounds electrically charged particles and exert forces on all other charged particles within the field, either attracting or repelling them.

• The electric field between two charges is similar to the gravitational field between two masses because both of them obey the inverse-square law with distance.
• The electric field can be visualized with a set of imaginary lines known as the electric field lines or lines of force.
• Electric field lines are originated from a positive charge and terminated on a negative charge.

Electric field lines

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What is Electric Charge?

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Electric charge is the property of a particle due to which it can apply force on another charged or uncharged particle.

• It is a scalar quantity and the SI unit is Coulomb (C).
• Dimensional formula of electric charge is [AT].
• The charge associated with a proton is known as a positive charge.
• The charge associated with an electron is known as a negative charge.

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Electric Field of a Point Charge Formula

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The electric field intensity (E) due to a point charge (Q) at any point in its electric field is defined as the electrostatic or Coulomb’s force (F) per unit of charge exerted on an infinitesimal positive test charge (q_o) at rest at that point.

Mathematically, the electric field intensity is given by

E = F/q_o

• It is a vector quantity.
• SI unit of electric field intensity is Newton per coulomb (NC^-1) or Volt per meter (Vm^-1).
• Dimensional formula of electric field intensity is [MLT^-3A^-1].

Electric field due to a point charge

According to Coulomb’s law, the force between two charges Q and q_o separated by distance r is given by

F = \(\frac{1}{4 \pi \epsilon_o} \frac{Qq_o}{r^2}\)

Where ∈_o is the absolute permittivity of free space.

Therefore, the electric field intensity is given by

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

Above expression represents the electric field due to a point at distance r.

Also Read:

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Derivation of Electric Field Due to Two Point Charges

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Consider two point charges q₁ and q₂ placed at A and B having position vector \(\vec{r_1}\) and \(\vec{r_2}\)

The force on q_o due to q₁ is given by

F₁ = \(\frac{1}{4 \pi \epsilon_0} \frac{q_0q_1}{|\vec{r}- \vec{r_1}|^3} (\vec{r} - \vec{r_1})\)

Force on q_o due to q₂ is given by

F₂ = \(\frac{1}{4 \pi \epsilon_0} \frac{q_0q_2}{|\vec{r}- \vec{r_2}|^3} (\vec{r} - \vec{r_2})\)

Electric field due to two pint charges

Net force acting on q_o placed at point P is given by

F = F₁ + F₂

⇒ F = \(\frac{1}{4 \pi \epsilon_0} \frac{q_0q_1}{|\vec{r}- \vec{r_1}|^3} (\vec{r} - \vec{r_1})\) + \(\frac{1}{4 \pi \epsilon_0} \frac{q_0q_2}{|\vec{r}- \vec{r_2}|^3} (\vec{r} - \vec{r_2})\)

⇒ F = \(\frac{q_0}{4 \pi \epsilon_0}\) \([\frac{q_1}{|\vec{r}- \vec{r_1}|^3} (\vec{r} - \vec{r_1}) + \frac{q_2}{|\vec{r}- \vec{r_1}|^3} (\vec{r} - \vec{r_2})]\)

Electric field intensity at point P, is given by

\(\vec{E}  = \frac{\vec{F}}{q_o}\)

⇒ \(\vec{E}\) = \(\frac{1}{4 \pi \epsilon_0}\) \([\frac{q_1}{|\vec{r}- \vec{r_1}|^3} (\vec{r} - \vec{r_1}) + \frac{q_2}{|\vec{r}- \vec{r_1}|^3} (\vec{r} - \vec{r_2})]\)

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Electric Field Due to a System or Group of Point Charges

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The electric field intensity at any point because of a system or group of charges is equivalent to the vector sum of electric field intensities because of individual charges at the same point. Currently, we would perform the vector sum of electric field intensities:

\(\vec{E} = \vec{E_1} + \vec{E_2} + \vec{E_3} + … + \vec{E_n}\)

\(\vec{E_1}\) = \(\frac{1}{4 \pi \epsilon_0}\) \(\sum _{i = 1} ^ {i=n} \frac{Q_i}{r_i^2} (\hat{ri})\)

Here, 

• r_i is the distance of the point P from the i^th charge Q_i 
• r_i is a unit vector directed to the point P from Q_i.

Let’s say charge Q₁, Q₂...Q_n is positioned in a vacuum at places r₁, r₂...., and r_? respectively.

Electric field due to multiple point charges

The net forces at P are the vector sum of forces because of individual charges, given by,

\(\vec{F}\) = \(\frac{1}{4 \pi \epsilon_0} q_0 \sum_{i=1}^{i=n} \frac{Q_i}{|\vec{r}- \vec{ri}|^3 ×|\vec{r}- \vec{ri}|}\)

As, \(\vec{E}  = \frac{\vec{F}}{q_o}\)

Therefore,

\(\vec{E}\) = \(\frac{1}{4 \pi \epsilon_0} \sum_{i=1}^{i=n} \frac{Q_i}{|\vec{r}- \vec{ri}|^3 ×|\vec{r}- \vec{ri}|}\)

Putting \(\frac{1}{4 \pi \epsilon_0}\) = k

\(\vec{E}\) = \(k\frac{Q_1}{r_1^2}+ k\frac{Q_2}{r_2^2} +.............+ k\frac{Q_n}{r_n^2}\)

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Electric Field due to Continous Distribution of Charge

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A system of closed spaced electric charges forms a continuous charge distribution. It can be on a line, on a surface, or on the entire volume.

To calculate the electric field due to charge distribution, the concept of charge density is introduced.

1. Linear charge density (λ): It is defined as the charge per unit length of the conductor. It is given by

λ = q/l

2. Surface charge density (σ): It is defined as the charge per unit area of the conductor. It is given by

σ = q/A

3. Volume charge density (ρ): It is defined as the charge per unit volume of the conductor. It is given by

ρ = q/V

Electric field intensity due to Linear charge distribution is given by

E = \(\frac{F}{q_o}\) = \(\frac{1}{4 \pi \epsilon_o} \int _L \frac{ \lambda dL}{r^2}\)

Electric field intensity due to Surface charge distribution is given by

E = \(\frac{F}{q_o}\) = \(\frac{1}{4 \pi \epsilon_o} \int _A \frac{ \sigma dA}{r^2}\)

Electric field intensity due to Volume charge distribution is given by

E = \(\frac{F}{q_o}\) = \(\frac{1}{4 \pi \epsilon_o} \int _V \frac{ \rho dV}{r^2}\)

Also Read:

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Things to Remember

• An electric field is a physical field encompassing electrically charged particles and exercising a force on all other charged particles in the field, either attracting or repulsing them.
• When an electric charge q₀ is positioned in close proximity to another charge Q, q₀ witnesses either an attractive or repulsive force.
• Electric field due to a point charge formula is given by E = \(\frac{F}{q_o}\) = \(\frac{1}{4 \pi \epsilon_o} \frac{Q}{r^2}\)
• Electric Field Due to two point charges is given by

\(\vec{E}\) = \(\frac{1}{4 \pi \epsilon_0}\) \([\frac{q_1}{|\vec{r}- \vec{r_1}|^3} (\vec{r} - \vec{r_1}) + \frac{q_2}{|\vec{r}- \vec{r_1}|^3} (\vec{r} - \vec{r_2})]\)

• A system of closed spaced electric charges forms a continuous charge distribution. It can be on a line, on a surface, or on the entire volume.
• Coulomb's law states that the magnitude of the electrostatic force of attraction or repulsion between two point charges is inversely proportional to the square of the distance between them and straight proportional to the magnitude of the charges’ product.