CBSE Physics for Class 12 | Electrostatics | Electrical Field

Electrical Field

A positive charge or a negative charge is said to create its field around itself.

If a charge $\displaystyle {{Q}_{1}}$ exerts a force on charge $\displaystyle {{Q}_{2}}$placed near it, it may be stated that since $\displaystyle {{Q}_{2}}$ is in the field of $\displaystyle {{Q}_{1}}$, it experiences some force, or it may also be said that since charge $\displaystyle {{Q}_{1}}$is inside the field of $\displaystyle {{Q}_{2}}$, it experience some force.

Thus space around a charge in which another charged particle experiences a force is said to have electrical field in it.

(1) Electric field intensity $\displaystyle (\vec{E})$:

The electric field intensity at any point is defined as the force experienced by a unit positive charge placed at that point. $\displaystyle \vec{E}= \frac{{\vec{F}}}{{{q}_{0}}}$

Where $\displaystyle {{q}_{0}}\to 0$so that presence of this charge may not affect the source charge Q and its electric field is not changed, therefore expression for electric field intensity can be better written as $\displaystyle \vec{E}=\underset{{{q}_{0}}\to 0}{\mathop{Lim}}\,\,\,\,\frac{{\vec{F}}}{{{q}_{0}}}$

(2) Unit and Dimensional formula of Electrical Field :

It’s S.I. unit –$\displaystyle \frac{Newton}{coulomb}=\frac{volt}{meter}=\frac{Joule}{coulomb\times meter}$ and

C.G.S. unit – Dyne/stat coulomb.

Dimension :
[$\displaystyle E$] =[$\displaystyle ML{{T}^{-3}}{{A}^{-1}}$]

(3) Direction of electric field :

Electric field (intensity) $\displaystyle \vec{E}$ is a vector quantity. Electric field due to a positive charge is always away from the charge and that due to a negative charge is always towards the charge

(4) Relation between electric force and electric field :

In an electric field $\displaystyle \vec{E}$ a charge (Q) experiences a force $\displaystyle F=QE$. If charge is positive then force is directed in the direction of field while if charge is negative force acts on it in the opposite direction of field

(5) Superposition of electric field (electric field at a point due to various charges) :

The resultant electric field at any point is equal to the vector sum of electric fields at that point due to various charges.

$\displaystyle \vec{E}={{\vec{E}}_{1}}+{{\vec{E}}_{2}}+{{\vec{E}}_{3}}+...$

The magnitude of the resultant of two electric fields is given by

$\displaystyle E=\sqrt{E_{1}^{2}+E_{2}^{2}+2{{E}_{1}}{{E}_{2}}\cos \theta }$ and the direction is given by $\displaystyle \tan \alpha =\frac{{{E}_{2}}\sin \theta }{{{E}_{1}}+{{E}_{2}}\cos \theta }$

(6) Electric field due to continuous distribution of charge :

A system of closely spaced electric charges forms a continuous charge distribution

Linear charge distribution

In this distribution charge distributed on a line.

For example : charge on a wire, charge on a ring etc. Relevant parameter is $\displaystyle \lambda$ which is called linear charge density i.e.,

$\displaystyle \lambda =\frac{\text{charge}}{\text{length}}$

$\displaystyle \lambda =\frac{Q}{2\pi R}$

Surface charge distribution

In this distribution charge distributed on the surface.

For example : Charge on a conducting sphere, charge on a sheet etc. Relevant parameter is $\displaystyle \sigma$which is called surface charge density i.e.,

$\displaystyle \sigma =\frac{\text{charge}}{\text{area}}$

$\displaystyle \sigma =\frac{Q}{4\pi {{R}^{2}}}$

Volume charge distribution

In this distribution charge distributed in the whole volume of the body.

For example : Non conducting charged sphere. Relevant parameter is $\displaystyle \rho$ which is called volume charge density i.e.

$\displaystyle \rho =\frac{\text{charge}}{\text{volume}}$

$\displaystyle \rho =\frac{Q}{\frac{4}{3}\pi {{R}^{3}}}$

To find the field of a continuous charge distribution, we divide the charge into infinitesimal charge elements.

Each infinitesimal charge element is then considered, as a point charge and electric field $\displaystyle \overrightarrow{dE}$ is determined due to this charge at given point.

The Net field at the given point is the summation of fields of all the elements. i.e., $\displaystyle \overrightarrow{E\,}=\int{\overrightarrow{dE}}$

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