Electric potential

Electric Potential:

An analogy. Consider two tubes A and B [in Fig.] Containing water and isolated from each other through a stop clock S. Amount of water in A is much less than in B while the level of water in A is higher than that in B. On opening stop clock it will be observed that water flows from higher level to lower one (i.e., from A to B) till the level becomes equal in two tubes. Thus, the factor that determines the direction of flow of water is the level of water and not the amount of water. Similarly in heat the factor that determines flow of heat from one body to the other is the temperature and not the amount of heat contained in the body. 

Fig. 1. Motion of water from a higher level to a lower level. 

     Correspondingly, in electricity, there is a quantity which determines the direction of flow of charge from one body to the other irrespective of the amounts of charges contained in the two bodies. This quantity is called electric potential of the body. Charges flows from a body at higher potential to that at a lower potential. 

Qualitative definition:

     Qualitatively, electric potential may be defined as the quantity which determines the direction of flow of charge between two bodies. 

In equation, V(rB) - V(rA) = - BA∫ E(r). dl     ..... (1), we have seen that negative line integral of electric field between any two points is equal to the difference between the values of scalar function V for two points rA and rB  . It is always convenient for us to choose this scalar function so that its value at infinite distance from the source of electric field is zero. Let the point rA  be removed to infinity. 

     ∴          V(rA) = 0

     So equation (i) can be written as

         

          V(r) = - ∞r∫ E(r). dl ...... (2)

     It may be noted that we have dropped the subscript 'B' here since it has become meaningless now. 

     Equation (2) defines the potential at a    point r. 

Quantitative definition:

     Electric potential at any point, is defined as the negative line integral of electric field from infinity to that point along any path. 

Potential difference:

     Using this concept, equation (2) defines the potential difference between any two points in the electric field. 

     Potential difference, between any two points, in an electric field is the negative line integral of electric field between them along any curve joining them together. 

Physical concept:

     Consider a small charge 'δq' being brought from infinity to a point r against an electric field of strength E(r'). Force F(r') acting on the charge δq while at a point r' is given by

           F(r') = δq E(r') 

     Since we have to push the charge against electric field we shall have to do work upon it. Thus, the potential energy of the charge will increase. Let this increase in potential energy be 'dW' when pushed through a small distance dl. 

               dW = - F(r'). dl

     Substituting for F(r')  dW = δq E(r'). dl

     If W is the work done in moving 'δq' from infinity to the point r', 

        W = W∞∫ dW = - δq ∞r∫ E(r'). dl ...... (3)

or

        W = δq. V(r)        [∵  - ∞r∫ E(r'). dl = V(r)]

 ∴   ∴        V(r) = W/δq                           ...... (4) 

If        δq = 1, V(r) = W. 

     Thus, potential, at any point, in an electric field, is defined as the work done in moving a unit positive charge from infinity to that point against the electric field along any path. 

Potential difference:

     If equation (3) is evaluated between two points rA  and rB , work done WAB  in that case is given by 

          V(rB) - V(rA) = WAB/δq

     In case of conservative field

      

           F = - dU/dx    or    dU = - Fdx

or

      Increase in potential energy = - (work done upon it) 

     If   δq = 1,      V(rB) - V(rA) = WAB.

     Potential difference, between any two points, in an electric field is defined as the work done in taking a unit positive charge from one point to the other against the electric field 

Units of Potential or Potential Difference:

(i) In S.I. units

       Unit of potential, in S.I. is volt (V) 

       If           δq = 1 C and W = 1 J

       Using equation (4),    

                  V(r) = 1 J/1 C = 1 volt. 

     Potential at any point, in an electric field, is said to be 1 volt if if 1 joule of work is done in moving a charge of 1 Coulomb between infinity to that point against the electric field along any path. 

(ii) In C.G.S. system:

      There are two types of units for potential in C.G.S. system. 

(a) e.s.u. of potential or statvolt

   1 e.s.u. of potential = 1 erg/1 e.s.u. of charge

     Potential, at any point is said to be 1 e.s.u. of potential or one statvolt of 1 erg of work is done in moving a charge of 1 e.s.u. between infinity to that point against the electric field along any path. 

(b) e.m.u. of potential or abvolt   

 

         1 e.m.u. of potential=1 erg/1 e.m.u. of charge

     Potential at any point, is said to be 1 e.m.u. of potential or 1 abvolt  if 1 erg of work is done in moving q charge of 1 e.m.u. between infinity to that point against the electric field along any path. 

Relation between volt and statvolt

          1 volt = 1 joule/1 coulomb

                     = 10⁷/3 × 10⁹ statcoulomb

∴        1 volt = 1/300 statvolt.

Relation between volt and abvolt

   

               1 volt = 1 joule/1 coulomb

                          = 10⁷ erɡ/1/10 abcoulomb               ∴        1 volt = 10⁸ abvolt

     It may be noted that abvolt is the smallest unit of potential while statvolt is the biɡɡest unit.

Relation between statvolt and abvolt

               1/300 statvolt = 10⁸ abvolt

    ∴         1 statvolt = 3 × 10¹⁰ abvolt

Electric potential at any point due to a single point charge

Electric potential at any point in an electric field is defined as the amount of work done in moving a unit positive charge between infinity to that point without any acceleration against the electric force. 

     If the source charge is positive, unit positive charge is moved from infinity to that point while in case of negative source charge, unit positive charge is taken from the given point to infinity. 

Case (i) Source charge situated at the origin

     Let the source charge q be situated at O. We intend to determine electric potential at Practice, distant r from O. Imagine moving unit positive charge from infinity towards People (Fig. 2). At any instant, during this process let the charge be at A, distant x from O. 

Fig. 2. Source charge at origin. 

     Let E (rightward arrow) be the electric intensity (force acting on a unit positive charge) at A. Small amount of work dW done to move the unit charge from A to B is 

           dW = E . dx = E dx cos 180° (E . dx is in right ward arrow) 

or          dW = - E dx 

     But      E = 1/4πε₀ × q/x²

     ∴      dW = - 1/4πε₀ × q/x² × dx          ... (5) 

     Net work done in moving the charge from infinity to P is

W = W0∫ dW = - q/4πε₀ r∞∫1/x² dx 

               = - q/4πε₀ r∞∫ x-2 dx

               = q/4πε₀ [x-2 +1/-2 + 1]r∞ 

               = - q/4πε₀ [x-1/-1]r∞

           W = - q/4πε₀ [-1/x]r∞ 

               = - q/4πε₀ [(-1/r) - (-1/∞)]

or      W = q/4πε₀ × 1/r

     According to the definition of electric potential V. 

                      V = W

∴                    V = q/4πε₀ × 1/r                ... (6) 

     Potential at any point is a scalar quantity (being the work done per unit charge). It is positive for a positive source and negative for a negative source charge. 

     Potential difference between two points A and B in the electric field will be

          VB - VA = q/4πε₀ × [1/| rA | - 1/r]

where rA and rB (both rA and rB is in rightward arrow) are the position vectors of A and B respectively. 

Case (ii) Source charge situated anywhere

Let r' (rightward arrow) be the position vector of source charge (Fig. 3).

Fig. 3. Source at any point. 

     According to triangle's law, 

                      SP = OP - OS  (SP, OP, OS are in rightward arrow) 

     ∴               SP = r - r'  (SP, r, r' are in rightward arrow) 

     Potential at P, in this case is given by

            V = 1/4πε₀ × q/|SP|

or     V = 1/4πε₀ × q/|r - r'|

     Potential difference between two points A and B will be 

            VB - VA = q/4πε₀ [1/| r - r' | - 1/| rA - r' |

where rA and rB (both rA and rB is in rightward arrow) are the position vectors of A and B respectively. 

Potential difference between two points

Potential difference between two points in an electric field is defined as the amount of work done in moving a unit positive charge from one point to the other, without any acceleration against the electric force. 

Case (i) Source charge situated at the origin

     Net work done in moving a  unit positive charge A to B (Fig. 3) can be obtained by integration equation (5) between the limits x = rA to x = rB. 

WAB = WAB0∫ dW = q/4πε₀ rBrA∫ 1/x² × dx

        = - q/4πε₀ [x-2 + 1/-2 + 1]rBrA

        = - q/4πε₀ [(- 1/rB) - (- 1/rA)]                       or        WAB = q/4πε₀ (1/rB - 1/rA) ... (7)

Fig. 4. Source charge at origin. 

     Equation (6), can be obtained from this by substituting rA = ∞ and rB = r. 

Case (ii) Source charge situated anywhere

     In case the source charge 'q' is situated at S [r' (rightward arrow)] (Fig. 5), the potential difference between two points A (rA) and B (rB) is given by

VB - VA = 1/4πε₀ [1/| SB | - 1/| SA | (SB and SA is in rightward arrow)

or    VB - VA = 1/4πε₀ [1/| rB - r' | - 1/| rA - r' |] (rB - r' and rA - r' is in rightward arrow)

Fig. 5. Source charge at anywhere. 

Potential Energy of a charge

We know that electric field exits at all the points around a source charge q' and every point is characterised with electric potential (work done in bringing a unit positive charge from infinity to that point) which is given by

             V(r) = 1/4πε₀ × q'/r

     If instead of bringing a unit charge , we bring a charge 'q' from infinity to that point, work done 'W' in doing so is given by

             W = q × V(r) (r is in rightward arrow) 

or         W = 1/4πε₀ × qq'/r

     This work done is termed as the potential energy of the charge q in the field of charge q'. 

     (i) If q and q' are of same nature, potential energy is positive. This is due to the repulsive force between them. 

    (ii) If q and q' are of opposite nature, potential energy is negative. This is due to the attractive force between them. 

     Thus, we can also say that the electric potential at any point in an electric field is numerically equal to the potential energy of a unit positive charge placed at that point. 

Important notes

(i) Electric potential is analogous to height in mechanics and temperature in beat.  (ii) Electric potential is a scalar quantity.  (iii) It is the negative line integral of electric field between infinity and the given point.  (iv) Electric potential at any point varies inversely as the distance of the point from source charge.  (v) Electric potential at any point due to a positive charge is positive while that due to a negative charge is negative.