Physics > Electric Charges and Fields

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  • Electric and magnetic forces determine the properties of atoms, molecules and bulk matter.
  • From simple experiments on frictional electricity, one can infer that there are two types of charges in nature; and that like charges repels 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.
  • 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.
  • 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  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.
  • Coulomb's Law: The mutual electrostatic force between two point charges  is proportional to the product  and inversely proportional to the square of the distance r21 separating them.

    Mathematically,

    F21 = force on q2 due to q1 =

    In SI units, the unit of charge is coulomb. The experimental value of the constant ε0 is

    The approximate value of k is

  • The ratio of electric force and gravitational force between a proton and an electron is
  • 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.
  • 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; 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.
  •  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.
  • Some of the important properties of field lines are:
  1. Field lines are continuous curves without any breaks.
  2. Two field lines cannot cross each other.
  3. Electrostatic field lines start at positive charges and end at negative charges —they cannot form closed loops.
  • An electric dipole is a pair of equal and opposite charges  separated by some distance 2a. Its dipole moment vector p has magnitude  and is in the direction of the dipole axis from
  • 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 from the centre:

    Dipole electric field on the axis at a distance r from the centre

    The  dependence of dipole electric fields should be noted in contrast to the 1/r 2 dependence of electric field due to a point charge.

  • In a uniform electric field E, a dipole experiences a torque τ given by  but experiences no net force.
  • The flux  of electric field E through a small area element ΔS is given by

    The vector area element ΔS is

     where ΔS is the magnitude of the area element and ˆn is normal to the area element, which can be considered planar for sufficiently small ΔS.

  • Gauss's law: The flux of electric field through any closed surface S is1/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:
  1. Thin infinitely long straight wire of uniform linear charge density λ. where r is the perpendicular distance of the point from the wire and ˆn 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 ˆn is a unit vector normal to the plane, outward on either side.

    (iii)             Thin spherical shell of uniform surface charge density σ.

E 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. 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 = 0                (r

   

Sample Examples

Question

If 109 electrons move out of a body to another body every second, how much time is required to get a total charge of 1 Con the other body?

Solution

In one second 109 electrons move out of the body. Therefore the charge given out in one second is

.

The time required to accumulate a charge of 1 C can then be estimated to be 1

.

Thus to collect a charge of one coulomb,

from a body from which 109 electrons move out every second, we will need approximately 200 years. One coulomb is, therefore, a very large unit for many practical purposes.

   

It is, however also important to know what is roughly the number of electrons contained in a piece of one cubic centimetre of a material.

A cubic piece of copper of side 1 cm contains about electrons.

 

Question

Consider three charges q1, q2, q3 each equal to q at the vertices of an equilateral triangle of side l. What is the force on a charge Q (with the same sign as q) placed at the centroid of the triangle?

Solution

In the given equilateral triangle ABC of sides of length l, if we draw a perpendicular AD to the side BC,

AD = AC cos 30º =   and the distance AO of the centroid O from A is (2/3) . By symmetry AO = BO = CO.

Force F1 on Q due to charge q at

Force F2 on Q due to charge q at

Force F3 on Q due to charge q at

The resultant of forces F2 and F3 is = along AO by the parallelogram law.

Therefore, the total force  0,

where rˆ is the unit vector along OA. It is clear also by symmetry that the three forces will sum to zero.

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