Electric Field and Electric Field Lines

 

A charged particle produces an electric field in the space surrounding it. When another charged particle is placed in this field, it experiences an electric force. Thus, the electric field provides a physical description of the influence of a charge on the space surrounding it.

The electric field at a point is defined as the region of space around a charge or a system of charges in which another charged particle experiences an electric force.

The concept of the electric field was developed by Michael Faraday. Electric field is a vector field, since it has both magnitude and direction. It is quantitatively described in terms of electric field intensity.

 

Intensity of Electric Field

The electric field intensity at a point is defined as the force experienced by a unit positive test charge placed at that point:

where is the force acting on the test charge .

The test charge is assumed to be a very small positive point charge, so that its presence does not appreciably disturb the charge distribution producing the electric field. More rigorously,

The direction of the electric field at a point is the direction of the force experienced by a positive test charge placed at that point.

Electric field intensity is a vector quantity and is usually denoted by . The term electric field is also commonly used to refer to electric field intensity.

The SI unit of electric field intensity is

It can also be expressed in , as will be discussed later.

The dimensional formula of electric field intensity is

Since

and

we obtain

Force on a Charged Particle in an Electric Field

If a charged particle carrying charge is placed in an electric field , the force acting on the particle is

For a positively charged particle , the force is in the same direction as the electric field:

For a negatively charged particle , the force is in the direction opposite to the electric field:

Thus, the direction of the force depends on the sign of the charge.

 

Direction of Electric Field

The electric field produced by a positive point charge is directed radially outward from the charge. Conversely, the electric field produced by a negative point charge is directed radially inward toward the charge.

 

 

 

a)              Electric Field Due to a Point Charge

Consider a point charge situated at a point , as shown in Fig. 1.9. We wish to determine the electric field intensity at a point , located at a distance from .

Let a small positive test charge be placed at .

 

Fig. 1.9: Electric field due to a point charge

 

According to Coulomb’s law, the force acting on the test charge due to is

where is the unit vector directed from towards the point .

By definition,

Therefore,

Hence,

For a positive point charge , the electric field at is directed radially outward.

If instead a point charge is placed at , then

Thus, for a point charge, the magnitude of electric field intensity is

Therefore,

for a given point charge.

The variation of electric field intensity with distance is shown in Fig. 1.10.

 

Electric Field in a Dielectric Medium

If the point charge is placed in a dielectric medium having relative permittivity (dielectric constant) , the permittivity of the medium is

Hence, the electric field is

In free space,

Therefore,

Since, for ordinary dielectrics,

we have

Thus, at the same distance from a given charge, the electric field intensity in a dielectric medium is reduced by a factor of compared with its value in free space.

 

 

Typical Values of Electric Fields

System / Situation	Typical electric field
Atmosphere near Earth's surface
Around domestic electric wires
Van de Graaff generator
Dielectric strength of air
X-ray tube

Note: The actual electric field can vary considerably depending on the specific system and operating conditions.

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b)             Electric Field Due to a System of Charges

The electric field due to a system of charges can be determined using the principle of superposition of electric fields.

According to this principle:

The electric field at a point due to a system of point charges is equal to the vector sum of the electric fields produced at that point by the individual charges, when each charge is considered independently.

Consider a system of point charges

as shown in Fig. Let the electric fields produced at point by these charges individually be

The net electric field at is therefore

or, in summation notation,

For the -th point charge , if its distance from is , then

Hence,

where is the unit vector directed from the -th charge towards the point .

Thus, electric fields obey the principle of superposition and must be added vectorially, not algebraically.

 

Electric Field Lines

 An electric field line is an imaginary line or curve drawn in a region of space where an electric field exists, such that the tangent to the field line at any point gives the direction of the electric field vector at that point.

 

Electric field lines have the following important properties:

(i) Direction of Electric Field Lines

Electric field lines originate from positive charges and terminate on negative charges.

For an isolated positive charge, the field lines extend radially outward and terminate at infinity. For an isolated negative charge, the field lines originate at infinity and terminate radially inward on the charge.

Thus,

• For : field lines are directed outward.
• For : field lines are directed inward.

(ii) Tangent Gives the Direction of Electric Field

In a charge-free region, electric field lines can be represented as continuous curves without breaks.

The tangent drawn to an electric field line at any point gives the direction of the electric field vector at that point.

Since

the electric field direction is the direction of the force acting on a unit positive test charge.

 

(iii) Number of Field Lines and Charge Magnitude

The number of electric field lines associated with a charge is taken to be proportional to the magnitude of the charge.

Thus, a charge of magnitude is represented by approximately twice as many field lines as a charge of magnitude , subject to the chosen convention for drawing the lines.

 

(iv) Field-Line Density Represents Electric Field Strength

The number of electric field lines crossing a unit area perpendicular to the field is proportional to the magnitude of the electric field at that location.

Therefore:

• Crowded field lines indicate a strong electric field.
• Widely spaced field lines indicate a weak electric field.

Hence, the electric field strength is represented by the density of field lines.

The strength of the electric field is proportional to the number of field lines crossing a unit area normal to the field.

If the field lines are more densely packed at point than at point , then

(v) Electric Field Lines Never Intersect

Two electric field lines can never cross each other.

If two field lines crossed at a point, there would be two different tangents at that point. This would imply two different directions of the electric field at the same point, which is impossible because the electric field vector at a given point has a unique direction.

 

(vi) Electrostatic Field Lines Are Not Closed Curves

In electrostatics, electric field lines cannot form closed loops.

They originate from positive charges and terminate on negative charges or extend to infinity. This is consistent with the fact that the electrostatic field is a conservative field.

Mathematically,

for an electrostatic field.

 

(vii) Attraction and Repulsion

The pattern of electric field lines provides a useful qualitative picture of electrostatic interactions.

The field-line pattern between unlike charges indicates their mutual attraction, while the pattern around like charges indicates their mutual repulsion.

Faraday also visualized field lines as having a longitudinal tension and a lateral repulsion, providing an intuitive mechanical picture of electric-field interactions. These are useful visualization concepts rather than literal mechanical properties of field lines.

 

(viii) Electric Field Lines and Equipotential Surfaces

Electric field lines are always perpendicular to equipotential surfaces.

This follows from the relation

which shows that the electric field is directed along the direction of maximum decrease of electric potential.

Since the surface of a charged conductor in electrostatic equilibrium is an equipotential surface, electric field lines are always normal (perpendicular) to the surface of the conductor.

 

Key Points

Property	Electric Field Lines
Direction	Tangent gives the direction of
Positive charge	Lines emerge outward
Negative charge	Lines terminate inward
Field strength	Greater line density means stronger field
Intersection	Field lines never intersect
Electrostatic field	Field lines are not closed loops
Equipotential surface	Field lines are perpendicular
Conductor in electrostatic equilibrium	Field lines are normal to the surface
Uniform field	Straight, parallel and equally spaced lines