ELECTRIC CHARGES AND FIELDS

ELECTRIC CHARGE

• Positive charge
• Negative charge

     

CONDUCTORS AND INSULATORS

METHODS OF CHARGING

1. By friction : By rubbing two suitable bodies, given in box one is charged by +ve and another by –ve charge in equal amount.

2. By conduction : Charging a neutral body by touching it with a charged body is called charging by conduction.

3. By induction : Charging a body without bringing it in contact with a charged body is called charging by induction.

PROPERTIES OF ELECTRIC CHARGE

• Similar charges repel and dissimilar charges attract each other.

• A charged body attracts light uncharged bodies, due to polarisation of uncharged body.

• Charge is conserved i.e., the charge can neither be created nor be destroyed but it may simply be transferred from one body to the other.

(Pair-annihilation process)

• Charge is unaffected by motion
  This is also called charge invariance with motion

• Quantisation of charge – A charge is an aggregate of small unit of charges, each unit being known as fundamental or elementary charge which is equal to e = 1.6 × 10–19 C. This principle states that charge on any body exists as integral multiple of electronic charge. i.e. q = ne where n is an integer.

COULOMB’S LAW

i.e., or

 

 

and = 9 × 109 Nm2 C–2.

Also in CGS system of unit.

 

 

Hence  

DIELECTRIC

• Polar dielectric
• Non-polar dielectric

DIFFERENCE BETWEEN ELECTROSTATIC FORCE AND GRAVITATIONAL FORCE

VECTOR FORM OF COULOMB’S LAW

SUPERPOSITION PRINCIPLE FOR DISCRETE CHARGE DISTRIBUTION: FORCE BETWEEN MULTIPLE CHARGES

FORCE FOR CONTINUOUS CHARGE DISTRIBUTION

Now the total force is calculated by integrating under proper limits.

i.e.,  

where is a variable unit vector which points from each dq, towards the location of charge q1 (where dq is a small charge element)

TYPES OF CHARGE DISTRIBUTION

• Volume charge distribution : If a charge, Q is uniformly distributed through a volume V, the charge per unit volume ρ (volume charge density) is defined by

; ρ has unit coulomb/m3.

• Surface charge distribution : If a charge Q is uniformly distributed on a surface of area A, the surface charge density , is defined by the following equation

σ has unit coulomb / m2

• Linear charge distribution : If a charge q is uniformly distributed along a line of length λ, the linear charge density λ, is defined by

, λ has unit coulomb/m.

• Fix the origin of the coordinate system on charge q.
• Draw the forces on q due to the surrounding charges considering one charge at a time.
• Resolve the force in x and y-axis respectively and find and
• The resultant force is and the direction is given byand the direction is given by

CALCULATION OF ELECTRIC FORCE IN SOME SITUATIONS

• Force on one charge due to two other charges

The direction of F is given by

• Force due to linear charge distribution

so, newton

1. When the distance between the two charges placed in vacuum or a medium is increased K-times then the force between them decreases K2-times. i.e., if F0 and F be the initial and final forces between them, then
2. When the distance between the two charges placed in vacuum or a medium is decreased K-times then the force between them increases K2-times. i.e., if Fo and F be the initial and final forces then F = K2Fo
3. When a medium of dielectric constant K is placed between the two charges then the force between them decreases by K-times. i.e., if Fo and F be the forces in vacuum and the medium respectively, then
4. When a medium of dielectric constant K between the charges is replaced by another medium of dielectric constant K’ then the force decreases or increases by (K/K’) times according as K’ is greater than K or K’ is less than K.

ELECTRIC FIELD

Electric field intensity due to a charge q at any position () from that charge is defined as

where is the force experienced by a small positive test charge q0 due to charge q.

If there are more charges responsible for the field, then

where are the electric field intensities due to charges q1, q2, q3…..respectively.

ELECTRIC LINES OF FORCE

1. The lines of force diverge out from a positive charge and converge at a negative charge. i.e. the lines of force are always directed from higher to lower potential.

           

2. The electric lines of force contract length wise indicating unlike charges attract each other and expand laterally indicating like charges repel each other.

3. The number of lines that originate from or terminate on a charge is proportional to the magnitude of charge.

i.e.,

4. Two electric lines of force never intersect each other.
5. They begin from positive charge and end on negative charge i.e., they do not make closed loop (while magnetic field lines form closed loop).

6. Where the electric lines of force are
   1. close together, the field is strong (see fig.1)
   2. far apart, the field is weak (see fig.2)

7. Electric lines of force generate or terminate at charges /surfaces at right angles.

ELECTRIC FIELD FOR CONTINUOUS CHARGE DISTRIBUTION

CALCULATION OF ELECTRIC FIELD INTENSITY FOR A DISTRIBUTION OF DIRECT AND CONTINUOUS CHARGE

1. Fix origin of the coordinate system where electric field intensity is to be found.
2. Draw the direction of electric field intensity due to the surrounding charges considering one charge at a time.
3. Resolve the electric field intensity in x and y-axis respectively and find ΣEx and ΣEy
4. The resultant intensity is and where θ is the angle between and x-axis.
5. To find the force acting on the charge placed at the origin, the formula F = qE is used.

ENERGY DENSITY
Energy in unit volume of electric field is called energy density and is given by
,

ELECTRIC FIELD DUE TO VARIOUS CHARGE DISTRIBUTION

• Electric Field due to an isolated point charge

• A circular ring of radius R with uniformly distributed charge

When x >> R,

E is max when . Also Emax

• A circular disc of radius R with uniformly distributed charge with surface charge density σ

• An infinite sheet of uniformly distributed charges with surface charge density σ

• A finite length of charge with linear charge density

and

For Infinite length of charge,

∴  and  

• Due to a spherical shell of uniformly distributed charges with surface charge density σ

• Due to a solid non conducting sphere of uniformly distributed charges with charge density ρ

 

 

  

• Due to a solid non-conducting cylinder with linear charge density λ

Eaxis = 0, ,

,

In above cases,

1. If the electric lines of force are parallel and equally spaced, the field is uniform.
2. If E0 and E be the electric field intensity at a point due to a point charge or a charge distribution in vacuum and in a medium of dielectric constant K then

3. If E and E’ be the electric field intensity at a point in the two media having dielectric constant K and K’ then

4. The electric field intensity at a point due to a ring with uniform charge distribution doesn’t depend upon the radius of the ring if the distance between the point and the centre of the ring is much greater than the radius of the ring. The ring simply behaves as a point charge.
5. The electric field intensity inside a hollow sphere is zero but has a finite value at the surface and outside it (; x being the distance of the point from the centre of the sphere).
6. The electric field intensity at a point outside a hollow sphere (or spherical shell) does not depend upon the radius of the sphere. It just behaves as a point charge.
7. The electric field intensity at the centre of a non-conducting solid sphere with uniform charge distribution is zero. At other points inside it, the electric field varies directly with the distance from the centre (i.e. E ∝ x; x being the distance of the point from the centre). On the surface, it is constant but varies inversely with the square of the distance from the centre (i.e.). Note that the field doesn’t depend on the radius of the sphere for a point outside it. It simply behaves as a point charge.
8. The electric field intensity at a point on the axis of non-conducting solid cylinder is zero. It varies directly with the distance from the axis inside it (i.e. E ∝ x). On the surface, it is constant and varies inversely with the distance from the axis for a point outside it (i.e. ).

MOTION OF A CHARGED PARTICLE IN AN ELECTRIC FIELD

Let a charged particle of mass m and charge q be placed in a uniform electric field, then electric force on the charge particle is

∴ acceleration, (constant)

• The velocity of the charged particle at time t is,

v = u + at = at = (Particle initially at rest) or

• Distance travelled by particle is
• Kinetic energy gained by particle,

Let and the particle enters the field with speed u along x-axis.

Acceleration along Y-axis,

…… (i)

i.e. y ∝ x2.

1. the particle will leave the field at a distance from its original path in the direction of field, given by
2. The particle will leave the region in the direction of the tangent drawn to the parabola at the point of escape.
3. The velocity of the particle at the point of escape is given b
   
   

4. The direction of the particle in which it leaves the field is given by

ELECTRIC DIPOLE

The torque τ on a dipole in uniform electric field as shown in figure is given by,

If and

Then 

and from θ1 → θ2,

i.e. U = –pE cosθ

where θ is the angle betweenand .

We can also write

ELECTRIC FIELD DUE TO AN ELECTRIC DIPOLE

• Along the axial line (or end-on position)

and are parallel

when x >> l

• Along equatorial line (or broadside on position)

when x >>l

When and are anti parallel then,

• At any point (from the dipole)

;

ELECTRIC FORCE BETWEEN TWO DIPOLES

when dipoles are placed coaxially

when dipoles are placed perpendicular to each other.

1. The dipole moment of a dipole has a direction from the negative charge to the positive charge.
2. If the separation between the charges of the dipole is increased (or decreased) K-times, the dipole moment increases (or decreases) by K-times.
3. The torque experienced by a dipole placed in a uniform electric field has value always lying between zero and pE, where p is the dipole moment and E, the uniform electric field. It varies directly with the separation between the charges of the dipole.
4. The work done in rotating a dipole in a uniform electric field varies from zero (minimum) to 2pE (maximum). Also, it varies directly with the separation between the charges of the dipole.
5. The potential energy of the dipole in a uniform electric field always lies between +pE and –pE.
6. The electric field intensity at a point due to an electric dipole varies inversely with the cube of the distance of the point from its centre if the distance is much greater than the length of the dipole.
7. The electric field at a point due to a small dipole in end-on position is double of its value in broad side-on position,

8. For a small dipole, the electric field tends from infinity at a point very close to the axis of the dipole to zero at a point at infinity.
9. The force between two dipoles increases (or decreases) by K4-times as the distance between them decreases (or increases) by K-times.
10. Time period of a dipole in uniform electric field is

ELECTRIC FLUX

For uniform electric field when the angle between area vector and electric field has the same value throughout the area,

1. The electric flux is a scalar although it is a product of two vectors and (because it is a scalar product of the two).
2. The electric flux has values lying between –EA and +EA, where E and A are the electric field and the area of cross-section of the surface.

GAUSS’S LAW

i.e.,

• The net electric flux through any closed surface depends only on the charge inside that surface. In the figures, the net flux through S is q1/εo, the net flux through S’ is (q2 +q3 )/εo and the net flux, through S” is zero.

  

• The net flux across surface A is zero

i.e.,

APPLICATIONS OF GAUSS’S LAW

• To determine electric field due to a point charge

so,

 

• To determine electric field due to a cylindrically symmetric charge distribution

1. The closed imaginary surfaces drawn around a charge are called Gaussian surfaces.
2. If the flux emerging out of a Gaussian surface is zero then it is not necessary that the intensity of electric field is zero.
3. In the Gauss’s law,

 is the resultant electric field due to all charges lying inside or outside the Gaussian surface, but Qin is the charge lying only inside the surface.

4. The net flux of the electric field through a closed surface due to all the charges lying inside or outside the surface is equal to the flux due to the charges only enclosed by the surface.
5. The electric flux through any closed surface does not depend on the dimensions of the surface but it depends only on the net charge enclosed by the surface.