Electric charges and fields

 The study of electric charges at rest is called Electrostatics.

1. Two Kinds of Electric Charges

When two bodies are rubbed together, they get oppositely charged. Experimental evidences show that there are two types of charges:

(i) Positive Charge: Positive charge is produced by the removal of electrons from a neutral body. That is, positive charge means deficiency of electrons.

(ii) Negative Charge: Negative charge is produced by giving electrons to a neutral body. That is, negative charge means excess of electrons on a neutral body.

SI unit of charge is coulomb (C).

2. Properties of Charges

(i) Conservation of Charge: The charge of an isolated system remains constant. This means that charge can neither be created nor destroyed; but it may simply be transferred from one body to another.

(ii) Additive Property: Total charge on an isolated system is equal to the algebraic sum of charges on individual bodies of the system. This is called additive property of charges. That is, if a system contains three charges, q1, q2, – q3, then total charge on system, Q= q1+ q2 – q3.

(iii) Quantisation of Charge: The total charge on a body is the integral multiple of fundamental charge‘e’

i.e., q = ± ne where n is an integer (n = 1, 2, 3,...).

(iv) Charge is unaffected by motion: The charge on a body remains unaffected of its velocity, i.e.,

Charge at rest = Charge in motion

(v) Like charges repel while unlike charges attract each other.

3. Coulomb’s Law in General Form

It states that the force of attraction or repulsion between two point charges is directly proportional to the product of magnitude of charges and inversely proportional to the square of distance between them. The direction of this force is along the line joining the two charges, i.e.,

F=k.q1q2r2

where k=14πεis constant of proportionality; ε is permittivity of medium between the charges. If ε0 is permittivity of free space and K the dielectric constant of medium, then ε=Kε0

F=14πε0Kq1q2r2

For free space K = 1, Therefore

∴ F=14πε0q1q2r2

Dielectric constant or Relative permittivity (K): The dielectric constant of a medium is defined as the ratio of permittivity of medium to the permittivity of free space, i.e., K = ε/ε0

Definition of coulomb: 1 coulomb charge is the charge which when placed at a distance of 1 metre from an equal and similar charge in vacuum (or air) will repel it with a force of 9 × 109 N.

4. Coulomb’s Law in Vector Form

Consider two like charges q1 and q2 located at points A and B in vacuum. The separation between the charges is r. As charges are like, they repel each other. Let F⃗ 21 be the force exerted on charge q2 by charge q1 and F⃗ 12 that exerted on charge q1 by charge q2. If r⃗ 21 is the position vector of q2 relative to q1 and rˆ21 is unit vector along A to B, then the force F⃗ 21 is along A to B and

F⃗ 21=14πɛ0q1q2r2rˆ21 ...(i)

But rˆ21=r⃗ 21r

F⃗ 21=14πε0q1q2r2.r⃗ 21r=14πε0q1q2r3r⃗ 21

Similarly if r⃗ 12is position vector of q1 relative to q2 and rˆ12 is unit vector from B to A, then

F⃗ 12=14πε0q1q2r2rˆ12=14πε0q1q2r3r⃗ 12...(ii)

Obviously r⃗ 12=–r⃗ 21, therefore equation (ii) becomes

∴ F⃗ 12=–14πε0q1q2r3r⃗ 21 ...(iii)

Comparing (i) and (iii), we get

F⃗ 21=–F⃗ 12

This means that the Coulomb’s force exerted on q2 by q1 is equal and opposite to the Coulomb’s force exerted on q1 by q2; in accordance with Newton’s third law.

Thus, Newton’s third law also holds good for electrical forces.

5. Principle of Superposition of Electric Charges

Coulomb’s law gives the force between two point charges. But if there are a number of interacting charges, then the force on a particular charge may be found by the principle of superposition. It states that

If the system contains a number of interacting charges, then the force on a given charge is equal to the vector sum of the forces exerted on it by all remaining charges.

The force between any two charges is not affected by the presence of other charges.

Suppose that a system of charges contains n charges ql, q2, q3, ... qn having position vectors r1→,r2→,r3→,…rn→ relative to origin O respectively. A point charge q is located at P having position vector r⃗  relative to O. The total force on q due to all n charges is to be found. If F⃗ 1,F⃗ 2,F⃗ 3,…F⃗ n, are the forces acting on q due to charges ql, q2, q3, ... qn respectively, then by the principle of superposition, the net force on q is

F⃗ =F⃗ 1+F⃗ 2+F⃗ 3+…+F⃗ n

If the force exerted due to charge qi on q is F⃗ i, then from Coulomb’s law in vector form

F⃗ i=14πε0qqi|r⃗ –r⃗ i|3(r⃗ –r⃗ i)

The total force on q due to all n charges may be expressed as

Here ∑ represents the vector-sum.

6. Continuous Charge Distribution

The electrostatic force due to a charge element dq at charge q0 situated at point P is

dF⃗ =14πε0q0dqR3R⃗ =14πε0q0dq|r⃗ –r′→|3(r⃗ –r′→)

The total force on q0 by the charged body is

F=14πε0q0∫dq(r⃗ –r′→)|r⃗ –r′→|3

For linear charge distribution, dq = λ dl, where λ is charge per unit length and integration is over the whole length of charge.

For surface charge distribution, dq = σ ds, where σ is charge per unit area and integration is for the whole surface of charge.

For volume charge distribution, dq = ρdV, where  ρis charge per unit volume and integration is for whole volume of charge.

Electric field

The electric field strength at any point in an electric field is a vector quantity whose magnitude is equal to the force acting on per unit positive test charge and the direction is along the direction of force.

If F⃗  is the force acting on infinitesimal positive test charge q0, then electric field strength, E⃗ =F⃗ q0. Therefore from definition, electric field can be given as

E⃗ =limq0→0F⃗ q0

The unit of electric field strength is newton/coulomb or volt/metre (abbreviated as N/C or V/m respectively).

(i) The electric field strength due to a point charge q at a distance r in magnitude form

|E|=|F|q0=14πε0qr2

In vector form, E⃗ =14πε0qr3r⃗ 

(ii) The electric field strength due to a system of discrete charge is

(iii) The electric field strength due to a continuous charge distribution is

E⃗ =14πε0∫dqr3r⃗ 

7. Electric field lines

The electric field line in an electric field is an imaginary smooth curve along which an isolated free positive test charge tends to move.

In terms of electric field lines the electric field strength is defined as follows:

The electric field strength at any point is defined as a vector quantity whose magnitude is measured by the number of electric field lines passing normally through per unit small area around that point and whose direction is along the tangent on field line drawn on that point.

Accordingly nearer the electric field lines, stronger is the electric field, and farther the electric field lines, weaker is the electric field. In figure, the electric field strength at A is greater than that at B.

Properties of electric field lines

(i) The electric field lines appear to start from positive charge and to terminate at negative charge.

(ii) The tangent drawn at any point on the field line gives the direction of electric field strength at that point and the direction of force acting on a positive charge at that point.

(iii) No two electric field lines can intersect each other because if they do so, then two tangents can be drawn at the point of intersection; which would mean two directions of electric field strength at one point and that is impossible.

(iv) The electric field lines have a tendency to contract lengthwise like a stretched elastic string and separate from each other laterally. The reason is that opposite charges attract and similar charges repel.

(v) The electric field lines do not form any closed loops.

(vi) The equidistant electric field lines represent uniform electric field while electric field lines at different separations represent non-uniform electric field (Figure).

8. Electric Dipole

A system containing two equal and opposite charges separated by a finite distance is called an electric dipole. Dipole moment of electric dipole having charges +q and – q at separation 2l is defined as the product of magnitude of one of the charges and shortest distance between them.

p⃗ =q.2⃗ l

It is a vector quantity, directed from – q to + q

[Remark: Net charge on an electric dipole is zero.]

9. Electric Field Due to a Short Dipole

(i) A point P on axis, E=14πε02pr3

(ii) At a point P´ on equatorial line,

E′=14πε0pr3

10. Electric Force and Torque on an Electric Dipole in a Uniform Electric Field

In a uniform electric field of strength E, the net electric force is zero; but a torque equal to pE sin θ acts on the dipole (where θ is the angle between directions of dipole moment p⃗  and electric field E⃗ ). This torque tends to align the dipole along the direction of electric field. Torque in vector form τ⃗ =p⃗ ×E⃗ .

11. Electric Flux

The total number of electric field lines crossing (or diverging) a surface normally is called electric flux.

Electric flux through surface element ∆S⃗ is∆φ=E⃗ .∆S⃗ =E∆S cos θ, where E⃗  is electric field strength.

Electric flux through entire closed surface is

φ=∫SE⃗ .dS⃗ 

SI unit of electric flux is volt-metre or Nm2C–1.

12. Gauss’s Theorem

It states that the total electric flux through a closed surface is equal to 1ε0times the net charge enclosed by the surface i.e., φ=∫SE⃗ .dS⃗ =1ε0∑q

13. Formulae for Electric Field Strength Calculated from Gauss’s Theorem

(a) Electric field due to infinitely long straight wire of charge per unit length λ at a distance r from the wire is

E=14πε02λr

(b) Electric field strength due to an infinite plane sheet of charge per unit area σ is

E=σ2ε0, independent of distance of point from the sheet.

(c) Electric field strength due to a uniformly charged thin spherical shell or conducting sphere of radius R having total charge q, at a distance r from centre is

(i) at external point Eext=14πε0qr2(r>R)

(ii) at surface point ES=14πε0qR2

(iii) at internal point Eint=0

(d) Electric field strength due to a uniformly charged non-conducting solid sphere of radius R at a distance r from centre

(i) at external point Eext=14πε0qr2(r>R)

(ii) at surface point ES=14πε0qR2

(iii) at internal point, Eint=14πε0qrR3(r<R)