Electric Potential

Definition: Potential at a point in a field is defined s the amount of work done in bringing a unit positive test charge; from infinite to that point along any arbitrary path. Electric potential defines as scalar quantity; it is denoted by V; V = W/q_o

Unit and dimensional formula: S.I. unit - Joule/couloumb= volt C.G.S. unit - State volt (e.s.u.); 1 volt  stat volt Dimension: [V] = [ML²T^-3A^-1-1-1]

Types of electric potential: It is defined as the nature of charge potential of charge potential is of two types:

(i) Positive potential: with respect to positive charge.

(ii) Negative potential: with respect to negative charge.

Potential of a system of point charges: Consider P is a point at which net electric potential is to be determined due to several charges. So net potential P

Electric potential due to a regular charge distribution: The potential due to a continuous charge distribution is the sum of potentials of all the infinitesimal charge elements in which the distribution may be divided i.e.,

Graphical representation of potential: When we move from a positive charge towards an equal negative charge along the line joining the two then initially potential decreases in magnitude and at centre converts into zero, but this potential is throughout positive because when we are nearer to positive charge, overall potential remain always negative but increases in value fig. (A). As one step from one charge to other when both charged particle are behave, the potential first decreases, at centre convert into minimum and then increases Fig. (B)

Potential difference

In an electric field difference between potential energy of two points A and B is defined as equal to the amount of work done (by external agent) in moving a unit positive charge from point A to point B.

i.e.,  in general W = Q.ΔV; ΔV =Potential difference through which charges Q moves.

To find potential at a point due to concentric sphere following guideline are to be considered

Guideline 1: Identity the point (P) at which potential sis to be determined.

Guideline 2: Start from inner most sphere, you should know where point (P) lies w.r.t. concerning sphere/shell (i.e. outside, at surface or inside)

Guideline 3: Then find the potential at the point (P) due to inner most sphere and then due to next and so on.

Guideline 4: Using the principle of superposition find net potential at required shell/sphere.

Standard Cases:

Case I: If two concentric conducting shells of radii r₁ and r₂ (r₂ > r₁) carrying uniformly distributed charges Q₁ and Q₂ respectively. What will be the potential of each shell

                        To find the solution of following guidelines are to be taken.

                        Here after following the above guideline potential at the surface of inner shell is

                        and potential at the surface of outer shell

Case II: The figure shows three conducting concentric shell of radii a, b and c (a < b < c) having chares  and  respectively what will be the potential of each shell

                        After given the guidelines defined above

Case III: The figure shows two concentric spheres having radii r₁ and r₂ respectively . If charge on inner sphere is +Q and outer sphere is earthed then calculate.

(a) The charge on the outer sphere

(b) Potential of the inner sphere

(i) Potential at the field layer of outer sphere

=>   Q′ = -Q

(ii) Potential of the inner sphere

Case IV: In the case III of outer sphere is given a charge +Q and inner sphere is earthed then

(a) What will be the charge on the inner sphere?

(b) What will be the potential of the outer sphere?

(i) In this case potential at the surface of inner sphere is zero, so if Q' is the charge induced on inner sphere then

(ii) Potential at the surface of outer sphere

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