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Electric Potential

Potential:

    It is the term in electrostatics analogous to the temperature in heat and thermodynamics. As temperature determines the direction of flow of heat, potential determines the direction of charge flow. As heat always flow from higher to lower temperature, charge also flows from higher to lower potential (negative charge flows from lower to higher potential). The potential are classified into three categories: positive, negative and zero potential.

    Potential of a body is determined by assuming earth to be at zero potential. If we connect any body to earth and positive charge flows from the body to earth then the body is at higher potential than earth or body is at positive potential. Similarly, if charge flows from the earth to body on connecting body to earth then body is at lower potential than earth or body is at negative potential.

    Mathematically potential at a point is defined as the amount of work done in bringing a unit positive charge from infinity to that point or potential is the work done per unit charge to bring the charge from infinity to that point,


Units :

    The units of electric potential are Joules / Coulomb (Common name is volt) or ergs/statcoulomb (common name statvolt)

Relation between Volt and Stat Volt :


Potential due to Monopole

:

    Consider a charge particle Q placed at a point A. We have to find the potential at any point P which is at a distance r from Q. Now as potential is the work done in bringing unit positive charge from infinity to that point. Imagine a small displacement dx in the path of the charge. Work done to move this small distance dx will be,



    Force between charge Q and unit charge at distance x is,


    Total work done in moving from infinity to r is,


Potential At a Point Due to An Electric Dipole

Consider an electric dipole consisting of charge q and –q kept at a distance 2a from each other. The electric potential is to be calculated at point P. Let AP = r1 and BP = r2 and angle POB = q where O is the mid point of dipole. Draw BN PO and AM PO. On = OB cos q = a cos q and OM = AO cos q = a cos q.

Thus, r1 = r + a cos q and r2 = r – acos q. And the potential at P is given by

V =

V =

V =

V =

V =

But, if acos q << r, then neglecting a cosq in comparison to r, we get potential as

V =

Special Cases: [1] If point P lies on the axial line of dipole, then q = 0

V =

[2] if point P lies on the equatorial line of dipole, then q = 90

V = =0

Potential Due to a Number of Charges:

    Potential at a point due to number of charges is the sum of potential at that point due to individual charges. For e.g. if we have number of charges Q1, Q2, ……, Qn and we have to find the potential at P which is at a distance of r1, r2, ……, rn from charges Q1, Q2, ……, Qn respectively. The potential at P is,

V  = V1 + V2 + V3 + – – – – – – – -+ Vn


    The potential is a scalar quantity, so simple addition is required without considering directions.

Potential Energy:

    It is the work done in bringing number of charges from infinity to their respective positions in the absence of any other charges.

Potential Energy For A System Of Two Charges :

    It is defined as the amount of work done in bringing the charges from infinity to their respective positions in the absence of any other charge. For eg. consider two charges q1 and q2 situated a distance ‘r‘ apart. To find the potential energy of the system of charges, first find the work done in bringing q1 from infinity to A, this is equal to zero, as in absence of any other charge force acting on it is zero. Similarly we find the work done in bringing q2 from infinity to B in the presence of q1. We imagine a small path ‘dx‘ in the path at a distance ‘x‘. Work done to move this small distance is,

dW = = F dx cos180º = – Fdx

where is the force acting on q2 at P.

    To find the total work done, integrate from ¥ to r,


    This work done is equal to potential energy and hence


Potential Energy for System of Three Charges:

    Consider three charges to be situated as shown in the figure. Work done in bringing q1 in absence of any other charge is zero. Similarly as in the above article work done in bringing q2 in the presence of q1 is,


    Work done in bringing q3 in presence of q1 and q2 is,


    Thus, total work done is,


    This total work done is the potential energy of the system. Hence for a system of n-charges potential energy in generalised form is,


Potential Difference:

    It is the difference of potential between any two points and is defined as the amount of work done in moving a unit positive charge from one point to another.

    Consider two points A and B with potential VA and VB. The work done in moving a charge particle from A to B is given by,


Electric field is Conservative in Nature:

    The force or field is said to be conservative in nature if work done in moving a particle in this field around any closed path is zero or work done in moving from one point to another is independent of the path followed. For example, if unit positive charge is to be moved from A to B via path ACB, then work done will be,


    Work done in moving the particle back from B to A via some other path ACB will be


    Thus, total work done in moving a particle around any closed loop BCBCA is


    Hence as total work done is zero therefore, electric field must be conservative in nature.

Electric Potential Difference and Electric Field Intensity:

    Consider any path AB in non uniform electric field and P be any point on this curve. We know that potential difference between A and B,


    If the point A is moved to infinity, the potential at any point is given by,


    Let Q be another point situaed very close to P at a very small distance ‘dl’ so that the field between P and Q is practically the same. In that case potential difference dV between these points is


    where ET is the tangential component of electric field in direction of dl or ET = dV/dl and the term is known as potential gradient. Hence if potential is constant in a certain region of space then electric field is zero.

Equipotential Surface

:An equipotential surface is that o which potential everywhere on the surface is same.

From the defination of electric potential surface

dV =

Thus, if the potential difference between two points on the surface is zero i.e. dV =0, it implies that no work is done in moving the charge between the two points dW =0

Work done to move a charge between two points is also given by

dW = =0

As q0, E and dx are not zero this implies that cos =0 or angle between electric field intensity and displacement vector is zero. Thus, electric field intensity is always perpendicular to the equipotential surface. Thus, no two equipotential surfaces can intersect, because if they do intersect there will be two direction of electric field intensity corresponding to two surfaces at the same point which is never possible.


Note: [a] If the electric field intensity is uniform, then the equipotential surfaces are planes with their surfaces perpendicular to the electric lines of force.

[b] If we have isolated point charge positive or negative then the equipotential surfaces are concentric spheres with their center coinciding with the position of charge.

[c] equipotential surfaces are crowded in the region of strong electric fields and are far apart in the weak fields.


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