Electric Charges and Fields
1. Electric and magnetic forces determine the properties of atoms, molecules and bulk matter.
2. From simple experiments on frictional electricity, one can infer that there are two types of charges in nature; and that like charges repel and unlike charges attract. By convention, the charge on a glass rod rubbed with silk is positive; that on a plastic rod rubbed with fur is then negative.
3. Conductors allow movement of electric charge through them, insulators do not. In metals, the mobile charges are electrons; in electrolytes both positive and negative ions are mobile.
4. Electric charge has three basic properties: quantisation, additivity and conservation.
Quantisation of electric charge means that total charge (q) of a body is always an integral multiple of a basic quantum of charge (e) i.e.,
q = n e, where n = 0, ±1, ±2, ±3, .... Proton and electron have charges +e, –e, respectively. For macroscopic charges for which n is a very large number, quantisation of charge can be ignored.
Additivity of electric charges means that the total charge of a system is the algebraic sum (i.e., the sum taking into account proper signs) of all individual charges in the system.
Conservation of electric charges means that the total charge of an isolated system remains unchanged with time. This means that when bodies are charged through friction, there is a transfer of electric charge from one body to another, but no creation or destruction of charge.
5. Coulomb’s Law: The mutual electrostatic force between two point charges q1 and q2 is proportional to the product q1q2 and inversely proportional to the square of the distance r21 separating them. Mathematically,
F21 = force on q2 due to
where
is a unit vector in the direction from q1 to q2 and k =
is the constant of proportionality.
In SI units, the unit of charge is coulomb. The experimental value of the constant ε0 is
ε0 = 8.854 × 10–12 C2 N–1 m–2
The approximate value of k is
k = 9 × 109 N m2 C–2
6. The ratio of electric force and gravitational force between a proton and an electron is
7. Superposition Principle: The principle is based on the property that the forces with which two charges attract or repel each other are not affected by the presence of a third (or more) additional charge(s). For an assembly of charges q1, q2, q3, ..., the force on any charge, say q1, is the vector sum of the force on q1 due to q2, the force on q1 due to q3, and so on. For each pair, the force is given by the Coulomb’s law for two charges stated earlier.
8. The electric field E at a point due to a charge configuration is the force on a small positive test charge q placed at the point divided by the magnitude of the charge. Electric field due to a point charge q has a magnitude |q|/4πε0r2; it is radially outwards from q, if q is positive, and radially inwards if q is negative. Like Coulomb force, electric field also satisfies superposition principle.
9. An electric field line is a curve drawn in such a way that the tangent at each point on the curve gives the direction of electric field at that point. The relative closeness of field lines indicates the relative strength of electric field at different points; they crowd near each other in regions of strong electric field and are far apart where the electric field is weak. In regions of constant electric field, the field lines are uniformly spaced parallel straight lines.
10. Some of the important properties of field lines are: (i) Field lines are continuous curves without any breaks. (ii) Two field lines cannot cross each other. (iii) Electrostatic field lines start at positive charges and end at negative charges —they cannot form closed loops.
11. An electric dipole is a pair of equal and opposite charges q and –q separated by some distance 2a. Its dipole moment vector p has magnitude 2qa and is in the direction of the dipole axis from –q to q.
12. Field of an electric dipole in its equatorial plane (i.e., the plane perpendicular to its axis and passing through its centre) at a distance r from the centre:
Dipole electric field on the axis at a distance r from the centre:
The 1/r3 dependence of dipole electric fields should be noted in contrast to the 1/r2 dependence of electric field due to a point charge.
13. In a uniform electric field E, a dipole experiences a torque
given by
= p × E
but experiences no net force.
14. The flux ∆φ of electric field E through a small area element ∆S is
given by
∆φ = E.∆S
The vector area element ∆S is
∆S = ∆S
where ∆S is the magnitude of the area element and
is normal to the area element, which can be considered planar for sufficiently small ∆S. For an area element of a closed surface,
is taken to be the direction of outward normal, by convention.
15. Gauss’s law: The flux of electric field through any closed surface S is 1/ε0 times the total charge enclosed by S. The law is especially useful in determining electric field E, when the source distribution has simple symmetry:
(i) Thin infinitely long straight wire of uniform linear charge density λ
where r is the perpendicular distance of the point from the wire and
is the radial unit vector in the plane normal to the wire passing through the point.
(ii) Infinite thin plane sheet of uniform surface charge density σ
where
is a unit vector normal to the plane, outward on either side.
(iii) Thin spherical shell of uniform surface charge density σ
E = 0 (r < R)
where r is the distance of the point from the centre of the shell and R the radius of the shell. q is the total charge of the shell: q = 4πR2σ.
The electric field outside the shell is as though the total charge is concentrated at the centre. The same result is true for a solid sphere of uniform volume charge density. The field is zero at all points inside the shell.