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 0so 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}}

CBSE Physics for Class 12 | Electrostatics | Coulomb’s Law

Coulomb’s Law.

If two stationary and point charges \displaystyle Q{}_{1} and \displaystyle Q{}_{2} are kept at a distance r, then it is found that force of attraction

 

or repulsion between them is \displaystyle F\propto \frac{Q{}_{1}Q{}_{{{2}_{{}}}}}{{{r}^{2}}} i.e., \displaystyle F=\frac{kQ{}_{1}Q{}_{2}}{{{r}^{2}}} ; (k = Proportionality constant)

(1) Dependence of k :

Constant k depends upon system of units and medium between the two charges.

(i) Effect of units

(a) In C.G.S. for air \displaystyle k=1, \displaystyle F=\frac{{{Q}_{1}}\,{{Q}_{2}}}{{{r}^{2}}} Dyne

(b) In S.I. for air \displaystyle k=\frac{1}{4\pi {{\varepsilon }_{0}}}=9\times {{10}^{9}}\frac{N-m{}^{2}}{C{}^{2}}, \displaystyle F=\frac{1}{4\pi {{\varepsilon }_{0}}}.\frac{{{Q}_{1}}{{Q}_{2}}}{{{r}^{2}}}Newton (1 Newton = 105
Dyne)

Note :

  • \displaystyle {{\varepsilon }_{0}}=Absolute permittivity of air or free space = \displaystyle 8.85\times {{10}^{-12}}\frac{{{C}^{2}}}{N-{{m}^{2}}}\displaystyle \left( =\frac{Farad}{m} \right). It’s Dimension is \displaystyle [M{{L}^{-3}}{{T}^{4}}{{A}^{2}}]
  • \displaystyle {{\varepsilon }_{0}}Relates with absolute magnetic permeability (\displaystyle {{\mu }_{0}}) and velocity of light (c) according to the following relation \displaystyle c=\frac{1}{\sqrt{{{\mu }_{0}}{{\varepsilon }_{0}}}}

(ii) Effect of medium

(a) When a dielectric medium is completely filled in between charges rearrangement of the charges inside the dielectric medium takes place and the force between the same two charges decreases by a factor of K known as dielectric constant or specific inductive capacity (SIC) of the medium, K is also called relative permittivity er of the medium (relative means with respect to free space).

Hence in the presence of medium \displaystyle {{F}_{m}}=\frac{{{F}_{air}}}{K}=\frac{1}{4\pi {{\varepsilon }_{0}}K}.\,\frac{{{Q}_{1}}{{Q}_{2}}}{{{r}^{2}}}

Here \displaystyle {{\varepsilon }_{0}}K={{\varepsilon }_{0}}\,{{\varepsilon }_{r}}=\varepsilon (permittivity of medium)     

(b) If a dielectric medium (dielectric constant K, thickness t) is partially filled between the charges then effective air separation between the charges becomes \displaystyle (r-t\,+t\sqrt{K})

Hence force \displaystyle F=\frac{1}{4\pi {{\varepsilon }_{0}}}\,\frac{{{Q}_{1}}{{Q}_{2}}}{{{(r-t+t\sqrt{K})}^{2}}}

(2) Vector form of coulomb’s law :

Vector form of Coulomb’s law is

\displaystyle {{\overrightarrow{F\,}}_{12}}=K.\frac{{{q}_{1}}{{q}_{2}}}{{{r}^{3}}}{{\overrightarrow{\,r}}_{12}}=K.\frac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}{{\hat{r}}_{12}},

where \displaystyle {{\hat{r}}_{12}} is the unit vector from first charge to second charge along the line joining the two charges.

(3) A comparative study of fundamental forces of nature

 

S.No. 

Force 

Nature and formula 

Range 

Relative

strength 

(i) 

Force of gravitation between two masses 

Attractive F = Gm1m2/r2, obey’s Newton’s third law of motion, it’s a conservative force

Long range (between planets and between electron and proton)

(ii) 

Electromagnetic force (for stationary and moving charges) 

Attractive as well as repulsive, obey’s Newton’s third law of motion, it’s a conservative force 

Long (upto few kelometers)

\displaystyle {{10}^{37}}

(iii) 

Nuclear force (between nucleons)

Exact expression is not known till date. However in some cases empirical formula \displaystyle {{U}_{0}}{{e}^{r/{{r}_{0}}}} can be utilized for nuclear potential energy \displaystyle {{U}_{0}} and \displaystyle {{r}_{0}} are constant.

Short (of the order of nuclear size 10–15
m)

1039

(strongest) 

(iv) 

Weak force (for processes like b decay)

Formula not known 

Short (upto 10–15m)

1024

 

Note :

  • Coulombs law is not valid for moving charges because moving charges produces magnetic field also.
  • Coulombs law is valid at a distance greater than \displaystyle {{10}^{-15}}m.
  • A charge \displaystyle {{Q}_{1}}exert some force on a second charge \displaystyle {{Q}_{2}}. If third charge \displaystyle {{Q}_{3}} is brought near, the force of \displaystyle {{Q}_{1}} exerted on \displaystyle {{Q}_{2}} remains unchanged.
  • Ratio of gravitational force and electrostatic force between (i) Two electrons is 10–43/1. (ii) Two protons is 10–36/1 (iii) One proton and one electron 10–39/1.
  • Decreasing order to fundamental forces \displaystyle {{F}_{Nuclear}}>{{F}_{Electromagnetic}}>{{F}_{Weak}}>{{F}_{Gravitational}}

(4) Principle of superposition :

According to the principle of super position, total force acting on a given charge due to number of charges is the vector sum of the individual forces acting on that charge due to all the charges.

Consider number of charge \displaystyle {{Q}_{1}},\displaystyle {{Q}_{2}},\displaystyle {{Q}_{3}}…are applying force on a charge Q

Net force on Q will be

\displaystyle {{\vec{F}}_{net}}={{\vec{F}}_{1}}+{{\vec{F}}_{2}}+..........+{{\vec{F}}_{n-1}}+{{\vec{F}}_{n}}