Electric Field: Definition, Formula & Electric Field Direction

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Electric Field Definition

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Electric field is the fundamental quantity that is defined as the electrostatic force per unit charge exerted on a small unit positive charge at that point. The direction of electric field lines are always from positive to negative point. The formula of the electric field is given by

\(E = \frac {F}{q}\)

Where

• E = Electric field intensity due to a point charge
• q = Electric charge
• F = Force experienced by the test charge due to the point charge

Let us suppose Q and q are two point charges that is separated by the distance r. Force F is exerted The charge Q exerts a force on charge q as per Coulomb’s Law, given by

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

To validate the fact that the force acts irrespective of whether the charge q is placed or not in the system, the concept of ‘Field’ was established. According to this, we say that the charge Q emits an electric field everywhere around it and when another charge q is brought into the system, it acts on it to produce a force.

Hence, the electric field due to a point charge Q at distance r is given by

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

From the above equation,

• It can be inferred that, if the magnitude of q is considered unity, i.e., 1 then the Force F becomes equal to the Electric Field E generated by the charge Q itself. 
• Thus, Electric Field E due to a charge Q at a point can be described as the force a unit positive charge would experience if placed at that point.
• Here, Q is called the source charge while any other charge q placed in the vicinity to test the effects of the source is called a test charge.
• Electric field intensity is a vector quantity and its SI unit is Newton per Coulomb (NC^-1).
• The dimensional formula of the electric field is [MLT^-3A^-1]

Electric Field

Electric Field Lines

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These are the straight or curved imaginary lines in space and the tangent to these field lines at any point gives the direction of the electric field at that point. Some properties of electric field lines are

• It begins from a positive charge and ends on a negative charge.
• It is an imaginary line such that tangents at any point on the field line give the direction of the electric field at that point.
• Electric field lines do not form a closed loop.
• In a charge-free region, it can be taken to be continuous curves without any break.
• Electric field lines never intersect each other.

Electric Field Lines Due to Different Charges

The electric field lines due to different charges are given below:

• The electric field lines due to a positive point charge are radially outward

• The electric field lines due to a negative point charge are radially inward

• Electric field lines due to a system of two similar positive charges are given by

Electric field lines due to two similar positive charges

• Electric field lines due to a system of two charges of equal and opposite charges or a dipole are given by

Electric field lines due to a system of two charges of equal and opposite charges

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Electric Field Lines due to System of Charges

In a system containing charges q₁, q_2, q₃,…, q_n, with position vector r₁, r₂, r₃,…, r_n, the net electric field intensity at a point is given as the vector sum of all the electric field strengths produced by the charges individually at that point.

Using the Superposition principle, electric field intensity at point P is given by

E(r) =E₁(r) + E₂(r) +…+ E_n

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Electric Field Lines due to a Dipole

An electric dipole is a pair of two equal and opposite charges separated by a certain distance. The product of the magnitude of either charge and the distance between the two charges (dipole length) is called the electric dipole moment.

An electric field due to a dipole is called a dipole field. Consider an electric dipole of dipole moment p, then the electric field intensity at distance r can be calculated on

• The axial line (end-on position) of the dipole, and
• The Equatorial line (Broad-side-on position) of the dipole

The formula for electric field intensity due to a short dipole on the axial line is given by

\(E = \frac {2p}{4\pi \epsilon_o r^3}\)

The formula for electric field intensity due to a short dipole on the equatorial line is given by

\(E = \frac {p}{4\pi \epsilon_o r^3}\)

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Electric Field due to Continous Distribution of Charge

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In this case, charges are distributed continuously over a length or a surface, or a volume. If we want to find the electric field with a surface that carries charges continuously over the surface, it is not possible to find the electric field due to each charged constituent. So in order to solve this problem we consider a small element and integrate it.

• If the total charge carried by an area element is equal to ΔQ, the surface charge density of the element is:

\(\sigma = \frac{\Delta Q}{\Delta s}\) (unit of σ is C/m²)

• Likewise, in the case of charge distribution along a line segment of length Δl, linear charge density is:

\(\lambda = \frac{\Delta Q}{\Delta l}\) (unit of λ is C/m)

• Similarly, if the total charge carried by a volume element is equal to ΔQ, the volume charge density of the element is:

\(\rho = \frac{\Delta Q}{\Delta V}\) (unit of ρ is C/m3)

Electric field intensity at distance r due to linear charge distribution is given by

\(\Delta E=\frac {1}{4\pi \epsilon_o}\frac {\lambda dl}{r^2}\)

Electric field intensity at distance r due to surface charge distribution is given by

\(\Delta E=\frac {1}{4\pi \epsilon_o}\frac {\sigma dS}{r^2}\)

Electric field intensity at distance r due to volume charge distribution is given by

\(\Delta E=\frac {1}{4\pi \epsilon_o}\frac {\rho dV}{r^2}\)

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Electric Field Lines due to Line Charge

Using Gauss law, the electric field due to line charge can be easily found. If we consider a line charge is in the form of a thin charged rod with linear charge density λ. 

Electric Field Due to Line Charge

In order to find electric intensity at point P at a perpendicular distance r from the rod,

• Let us assume a right circular closed cylinder of radius r and length l along with an infinitely long line of charge as its axis.
• As all points are at the same distance from the line charge, therefore the magnitude of the electric field intensity at every point on the curved surface of the Gaussian surface is the same.
• Thus, the curved surface of the cylinder will be: 

Gaussian surface

Electric Field Lines due to Ring

The electric field of a ring of charge on the axis of the ring can be found by superposing the point charge fields of infinitesimal charge elements. Furthermore, in order to calculate the electric field of a charged disc, the ring field can be used as an element. 

Here, we determine the field at point P on the axis of the ring. The field dE as a charge element dq is shown, and the total field is the superposition of all such fields because of all charge elements around the ring. While, the sum of the perpendicular field to zero, the differential x-component of the field is:

Electric Field Due to Ring

Electric Field due to a Uniformly Charged Sphere

Let’s consider σ as the uniform surface charge density of a sphere of radius R.

Next, find out the electric field intensity at a point P outside or inside the shell.

Field Outside the Shell

We have to find the electric field intensity at a point P outside the spherical shell such that, OP = r. 

• Here we take the Gaussian surface as a sphere of radius r.
• Then the electric field intensity is the same at every point of the Gaussian surface directed radially outwards. 
• Electric intensity at any point outside the spherical shell is such as if the entire charge is concentrated at the center of the shell.

The field at the Surface of the Shell

Here we have, r = R

Thus,

E = q / (4 π R² ε_o)

If σ C/m² is the charge density on the shell,

Then,

q=4 π R². σ

Thus,

E = (4 π R². σ ) / (4 π R². ε_o) = σ /ε₀

Field Inside the Shell

If the point P lies inside the spherical shell then the Gaussian surface is a surface of a sphere of radius r.

Since there’s no charge inside the spherical shell, the Gaussian surface encloses no charge. That is q = 0.

Thus, E = 0

Hence the field inside the spherical shell is always zero.

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Applications of Electric Field

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If the force on a charge can be determined directly using Coulomb’s Law and Superposition Principle, what is the need for Electric Field? In Electrostatics, the concept of an electric field eases the quantification process but is not necessary, then why introduce the notion of an Electric Field?

The following few reasons can be established for describing the significance of electric field:

• It provides a way of characterizing the electrical environment of the system of charges.
• It tells the force experienced by a unit positive test charge placed at a particular point.
• It is a vector field that refers to a quantity that is defined at every point in space and is variable in nature i.e., it varies from point to point.
• It tells the magnitude and direction of the Electric Force.
• The concept of Electric Field becomes very important and useful when time-varying electromagnetic phenomena come into the picture i.e., when we consider force between two charges separated by a distance in an accelerated motion.

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Things to Remember

• Electric Charge is associated with the matter due to which it produces and experiences electric and magnetic effects.
• The transference of electrons is the cause of frictional electricity.
• The total charge of an isolated system is always conserved, i.e. initial and final charge of the system will be the same.
• Coulomb’s Law states that the electrostatic force of interaction or repulsion acts between two stationary point charges.
• Electrostatic forces (Colombian forces) are conservative forces.