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Electric Field Gradient

Potential Gradient: Relation between Potential Gradient and Intensity of Electric FieldLet us consider the electric field E along the X-axis due to a point-charge + q at a point O (Fig. 12). Suppose A and B are two points distant x and x + dx from O, where dx is vanishingly small. Let V and V - dV be electric potentials at A and B respectively.Suppose a small positive test-charge q0 is moved / in the electric field from point B to point A. 


A force F will act on q0 in the direction of the field, whereF = qQE ...(i)Therefore, in moving q0 from B to A, an external agent will have to work against the force F. If this work be dW, thendW = F (- dx),where (- dx) is the displacement from B to A. Substituting the value of F from eq. (i), we getdW = —q0EdxdW r* j r-sor — = —Edx. ...(ii)%But, by the definition of potential difference, we havedW— = V-{V-dV) = dV. ...(iii)%Comparing eq. (ii) and (iii), we get-Edx = dVor E = - —.dxThe quantity dV/dx is the rate of change of potential with distance, and is known as 'potential gradient'. 


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Thus, the electric field intensity at a point in an electric field in a given direction is equal to the negative potential gradient in that direction. The negative sign signifies that the potential decreases in the direction of electric field.It is evident from the above relation that the electric field £ can also be expressed in the unit 'volt/meter' (V nf ')■ ThusINC"' = 1 Vm"'.r A . . ... dimensions of p.d.Dimensions of potential-gradient = —-:-——f---dimensions of displacement= [ML2T-3A"'] =[MLT-3A-,] m two metallic plates having positive and negative charges. 


If the plates are long and the distance between them is small, then the electric field produced between them will be uniform and directed from the positive plate to the negative plate. If the potentials of the plates be V\ and V2 volt and the distance between them be d meter, then the electric field E between the plates will be given byE = 1 . 2 volt/meter. dTrajectory of a Charged Particle in a Uniform Electric FieldThe motion of a charged particle in an electric field is same as that of a projectile in a gravitational field. Suppose two parallel metallic plates placed at some distance apart have opposite charges. Except near the edges, the electric field between the plates is uniform. 


If the upper plate is positively-charged and the lower is negatively-charged, then the electric field E will be directed downward in the plane of the paper (Fig. 14). The co-ordinate axes OX and OY are as shown in the figure.Suppose an electron (charge - e) moving along the X-axis enters the electric field with velocity v . 


The field is along the negative direction of Y-axis (downward). Hence no force acts on the electron along X-axis, but a force F acts along Y-axis, given byF = eE.The acceleration of the electron due to this force isF eE ...a = — = —, ...(l)m mwhere m is the mass of electron. The force F (or the acceleration a) is directed upward (towards the positive plate), as the electron is negatively charged. As a result of this force (or acceleration), the electron is deflected upward from its original path.Let us consider the motions of the electron along X- and Y-axes separately. Along the X-axis, the electron continues to move with its initial velocity v, because there is no acceleration along this axis. 


Therefore, the distance travelled in t second along the X-axis is given byx = vt. ...(ii)The initial velocity of the electron along the Y-axis is zero but along this direction the electron hasacceleration a . Therefore, the velocity of the electron continues to increase along the + Y-axis. Thedistance travelled in this direction in t second is given by1 2 y = -at .Substituting the value of a from eq. (i), we have1 ( r\1 eE 2y = ~ — t . ...(in)Z rnSubstituting the value of t from eq. (ii) in eq. (iii), we get_ 1 le_E (xy ~ 2 m v\ J \ JeE 2or y ■= --- x .2m v2This equation is of the form y - cx and represents a parabola. 


Hence, the trajectory of an electron in an electric-field is parabolic.After leaving the electric field, the electrical force acting on the electron vanishes and the electron moves in a straight line with velocity v' (> v).The electron enters the space between the plates with horizontal velocity v. Since no force is acting on the electron in the horizontal direction, its horizontal velocity v remains unchanged, and it covers a distance / (length of the plates) in time t. Clearly,x = I.Then, we can write_ eEl2 _2 mv2I 2If K be the kinetic energy of the electron entering the space between the plates, then K = - mv .eEl2 .. .•• y = TF- -(,v)Equipotential SurfacesAny surface over which the electric potential is same everywhere is called an equipotential surface.An equipotential surface may be the surface of a charged body or simply a surface in space. For example, the surface of a conductor is an equipotential surface. Equipotential surfaces can be drawn through a space in which there is an electric field. As an example, let us consider the electric field of an isolated point-charge +q. 


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The potential at a distance r from the charge isv 1 i. 4neo rA sphere of radius r with centre at + q is, therefore, an equipotential surface of potential q/4nto r. If fact, all spheres centred on + q are equipotential surfaces, whose potentials are inversely proportional to r (Fig. 15).The important properties of equipotential surfaces are : (i) No work is done in moving a charge between two points on an equipotential surface. This is so, because the potential difference between any two points on the surface is zero.(ii) The electric field and hence lines of force, are every where at right angles to the equipotential surface. This is so, because there is no potential gradient along any direction parallel to the surface, and so no electric field parallel to the surface (E = - dV/dr = 0). 


This means that the electric field E , and hence the lines of force, are always at right angles to the equipotential surface (only then the component of E parallel to the surface would be zero).In Fig. 15. the lines of force are radial and hence perpendicular to the equipotential surfaces.(iii) In a family of equipotential surfaces, the surfaces are closer together where the electric field is stronger, and farther apart where the field is weaker. 


This follows from the relation E = - dV/dr, or for the same potential-change dV, we havedr °c l/E ;that is, the spacing between the equipotential surfaces will be less where £ is strong, and vice-versa. Thus, equipotential surfaces can be used to give a general description of electric field in a certain region of space.(iv) No two equipotential surfaces can intersect each other. An equipotential surface is normal to the electric field. If two equipotential surfaces intersect each other then at the point of intersection there will be two directions of electric field, which is impossible.Both lines of force and equipotential surfaces can be used to depict electric field in space. The advantage of using equipotential surfaces over the lines of force is that they give a visual picture of both the magnitude and the direction of the electric field.