Relation Between Electric Field And Electric Potential
1 Considering a charge +q located at origin O, as shown.
First we find the work done in moving a test charge qo for displacement dr from point A to B having potentials V and V+dV respectively, as shown. Since dr is very small so we can assume that electric field E is nearly same at both points A and B. 2 The work done to move the test charge from A to B is, 0 W = F . dr= F dr cos180 = -qo E dr -----(1) Also the work in moving the test charge from A to B is, W = charge x potential difference
W = qo (VB - VA) = qo dV -----(2) Equating above two equations, we get
-qo E dr = qo dV 푑푉 Or, E = − (V/m) 푑푟 Hence the electric field at any point is equal to the negative of the potential gradient at that point. Negative sign shows that the direction of the electric field is in the 3 direction of decreasing potential. 4 5 6 Equipotential Surfaces & Their Properties
7 EQUIPOTENTIAL SURFACES Definition A surface that has same electric potential at every point on it, is called equipotential surface. It can be the surface of conducting body or an imaginary surface in electric field. It can also be defined as, a surface formed by taking locus of all those points which are at same potential. 8 Properties of Equipotential Surfaces
9 1. No work is done in moving a test charge over an equipotentialsurface. Reason:
In figure, since potential at points A and B are same so potential difference between these points will be zero.
So work done WAB = Q0 ( VB - VA) = 0 10 2. Two equipotential surfaces never intersect each other.
Reason: We give its reason by contradictory statement that, If they intersect, then there will be two values of electric potential at the point of intersection, which isimpossible. 11 3. Equipotentialsurfacesare closer together in region of strong electric field and farther apart in the region of weak field. Reason: We know that electric field at any point is equal to the negative of potential gradient at that point.
For constant dV; Thus, as we increase electric field, the spacing between surfaces will decrease, and vice versa. 12 4. The electric field is always normal to the equipotentialsurfaceat everypoint. Reason: We give its reason by contradictory statement that, if electric field is not normal then it would have non zero component along the surface. So a work have to be done to move a test charge along its surface. But that work done is always zero so electric field is always normal tothe equipotentialsurface. 13 EQUIPOTENTIAL SURFACES OF VARIOUS CHARGE SYSTEMS
14 1. Equipotential surface of a positive point charge:
15 2. Equipotential surface of a negativepoint charge:
16 3. Equipotential surface of an electric dipole (or two equal and oppositecharges):
17 4. Equipotential surface of two equal positive charges:
18 4. Equipotential surfaces for a uniform electric field:
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