Electric Field|What is  Electric Field ?|Definition

Electric Field : The electrostatic force acts between two charged bodies even without any direct contact between them. The nature of this action- at-distance force can be understood by introducing the concept of electric field.

Consider a charged body carrying a positive charge q placed at point O. It is assumed that the charge q produces an electrical environment in the surrounding space, called electric field.

To test the existence of electric field at any point P, we simply place a small positive charge q, called the test charge at the point P. If a force F is exerted on the test charge, then we say that an electric field E exists at the point P. The charge g is called the source charge as it produces the field E.

 Electric field

An electric field is said to exist at a point if a force of electrical origin is exerted on a stationary charged body placed at that point.

Quantitatively, the electric field or the electric intensity or the electric field strength point is defined as the force experienced by a unit positive test charge placed at that point, without disturbing the position of source charge.

Mathematical Expression of Electric field

Suppose we have a point charge Q at the position ‘ and another small point charge a placed at the position F with respect to some origin O (Fig 1). We are to find the force on q due to Q. Here Q is called the source charge and q is called test charge. From Coulomb’s law this force is

\displaystyle \vec{F}=\frac{1}{4\pi \epsilon _{0}}\cdot \frac{Q\cdot q}{R^{2}}\widehat{R}=q\cdot \vec{R}\ ....(1)

,

where

\displaystyle \vec{E}=\frac{1}{4\pi \epsilon _{0}}\cdot \frac{Q}{R^{2}}\widehat{R}\ ......(2)

R is the distance between Q and and q, \displaystyle \widehat{R}

  = unit vector in the direction from Q to q.

Obviously E = F/q, i.e., the force per unit charge placed at the point P. E is a function of the position vector R of P with respect to the source charge Q. It depends on Q but is independent of the test charge q.

Thus in the region of space surrounding the source charge we can uniquely specify a vector physical quantity E at every point. This means that E defines a vector field.

Thus the force F on q due to Q may be considered as arising in two steps.

(i) The charge Q sets up an electrical environment, called the E, in the surrounding space.

(ii) When a charge q is placed at any point without disturbing the position of Q it experiences a force, which equals the charge q multiplied by E at that point.

From Equation (2) electric field at a point in space due to a charge Q may be defined the force on a unit positive charge placed at that point.

The unit of E is N.C-. A unit charge may be high enough to disturb the configuration of the charge producing the field.

A way out of this difficulty is to choose the test charge q negligibly small. Then we may define the electric field at a point as the limiting force per unit charge placed that point, i.e.,

\displaystyle \vec{E}=\lim_{q\rightarrow 0}\frac{\vec{F}}{q}

.

Note that \displaystyle q\rightarrow 0

  is an idealization because \displaystyle q\rightarrow 0  contradicts the quantization of charge. In case of immobile source charges we need not require this idealization. Our definition of electric field in Equation (2) assumes source of the field is a point charge Q.

If the source is a set of discrete point charges \displaystyle q_{1},q_{2},.....,q_{N}

whose position vectors with respect to the origin O of some coordinate system are respectively \displaystyle \vec{r_{1}},\vec{r_{2}},....,\vec{r_{N}}  then the electric field at any point \displaystyle P(\vec{r})  can be calculated by using the principle of superposition as,

\displaystyle \vec{E}=\vec{E_{1}}+\vec{E_{2}}+\cdot \cdot \cdot \cdot +\vec{E_{N}}=\sum_{j=1}^{N}\vec{E_{j}}=\frac{1}{4\pi \epsilon _{0}}\sum_{j=1}^{N}\frac{q_{j}}{R_{j}^{2}}\widehat{R_{j}}\ ....(3)

.

where R, is the distance of the point observation P from the location of the point charge q; and R; is the unit vector in the direction from q; to P. Equation (3) can also be put in the following form :

\displaystyle \vec{E}=\sum_{j=1}^{N}\vec{E_{j}}=\frac{1}{4\pi \epsilon _{0}}\sum_{j=1}^{N}\frac{q_{j}(\vec{r}-\vec{r_{j}})}{\left |\vec{r}-\vec{r_{j}} \right |^{3}}

.

Units and dimensions

As the E is force per unit charge, so its SI unit is newton per coulomb (\displaystyle NC^{-1}

). It is equivalent to volt per metre (\displaystyle Vm^{-1} ).

The dimensions for \displaystyle \vec{E}

 can be determined as follows :

\displaystyle [E]=\frac{Force}{Charge}=\frac{MLT^{-2}}{C}

\displaystyle =\frac{MLT^{-2}}{A.T}=[MLT^{-3}A^{-1}]\ \ [\because 1A=\frac{1C}{1s}]

 Physical significance

The force experienced by the test charge \displaystyle q_{0}

  is different at different points. So \displaystyle \vec{E}  also varies from point to point. In general, \displaystyle \vec{E}  is not a single vector but a set of infinite vectors.

Each point \displaystyle \vec{r}

  is associated with a unique vector \displaystyle \vec{E(r)} . So electric field is an example of vector field.

By knowing \displaystyle \vec{E(r)}

 at any point, we can determine the force on a charge placed at that point. The Coulomb force on a charge g, due to a source charge \displaystyle q_{0} may be treated as two stage process :

(i) The source charge q produces a definite field \displaystyle \vec{E(r)}

at every point \displaystyle \vec{r} .

(ii) The value of \displaystyle \vec{E(r)}

at any point \displaystyle \vec{r}  determines the force on charge q, at that point. This force is,

\displaystyle \vec{F}=q_{0}\vec{E(r)}

.

Electrostatic force= Charge x Electric field

Thus an electric field plays an intermediary role in the forces between two charges :

Charge \displaystyle \rightleftharpoons

Electric field \displaystyle \rightleftharpoons Charge.

It is in this sense that the concept of electric field is useful. Electric field is a characteristic of the system of ric charges and is independent of the test charge that we place at a point to determine the field.

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