Title 17.Electrostatic Potential and Capacitance-F.pdf ✅

1 | P a g e NEET-2022 Ultimate Crash Course PHYSICS Electrostatic Potential and Capacitance
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3 | P a g e POINTS TO REMEMBER 1. Potential at a point is, in fact, pd between the potential at that point and the potential at infinity. 2. Since U =q0V, we may define the potential energy of a charge q0at any point in the field to be equal to the amount of work done in bringing the charge q0 from infinity to that point. 3. Both potential and potential energy are scalar quantities, because W and q0are scalars. 4. In the discussion above, we have assumed that the charged particle, which is moving is in equilibrium. If no external force is applied to maintain this equilibrium, the charged particle will change its kinetic energy as well as its potential energy as it moves from point to point. Thus, if a particle of mass m and charge q0has potential energy UA and kinetic energy KA at point A and UB and KB at B, the conservation of energy requires (in the absence of any external force acting on the charge), Of course, this is true only if the electric field is conservative, and we shall show this to be so in. 5. There is another way to interpret eqn. (2). Let | WAB be the work done by the electric force in displacing charge q0 from A to B. Since external force is equal and opposite to the electrostatic force, | | W W W AB AB BA = − = Thus, | 0 0 AB BA B A W W V V q q − = 6. At any location r let us express potential difference dV across a displacement dl that is not necessarily parallel to E . If figure, the angle between E and dl is  . Clearly, dV E dl Edl E dl = − = − = − ( cos cos .   ) Further, if E cos  is denoted by El (component of E along dl ), then / l l dV E dl or E dV dl = − = − ... (4) Eqn. (4) is the generalization of one dimensional and states that : The change of potential per unit displacement in a given direction is equal to the negative of the component of the electric field in that direction. 7. If the electric field has only one component Ex, then . E dl E dx = x . Thus, x dV E dx = − or / E dV dx x − ..(5) 8. In general, the electric potential is a function of all three spatial coordinates. If V (r) is given in terms of rectangular coordinates, the electric field components Ex, Ey, and Ez can readily be found from V(x, y, z) : , , x y z V V V E E E x y z    = = =    ----------------(6) In these expressions, the derivatives are called partial derivatives. 4. In a uniform field, E is constant in magnitude and direction at all points, hence dV/dr is constant, i.e., the
4 | P a g e potential changes steadily with distance, i.e., V E r = − . The field near the centre of two parallel metal plates is uniform and if this is created by a pd V between plates of separation r, then E = -V/r, where E is the field strength at any point in the uniform region. 5. Since dV E dx = − and for a point charge, 2 , e q E k r = e 2 q dV Edr k dr r = − = − Or 2 1 r r e e e q q V k dr k q k r r r     = − = =      6. As e 2 q E k r = and , e q V k r = 2 / 1 / e e E k q r V k q r r = = or V E r = From here it follows that the unit of E is volt/metre (V/m). Further, the two units of E (i.e., N/C and V/m) are the same as V J C Nm C N / / m m m C = = = (as J = N and V = J/C) The unit of electric flux, i.e., 2 Nm C/ can also be expressed as: 2 / Nm J Nm C m m Vm C C     = = =         7. Potential due to an isolated positive charge is positive and that due to an isolated negative charge is negative. 8. Potential is a scalar quantity. 9. Since the potential at a point in the electric field of a charge q depends upon r (the position of the point), we say that the potential is a single valued scalar function of the position of the point. 10. Potentials at equal distances from a point charge are the same. This means that the electric potential due to a single charge is spherically symmetric. 11. Since all points lying on the surface of a spherical conductor are at the same potential, no work is done to move a charge from one point on the surface to another point on the surface. 12. 0 W W Edl AB BA + = =  is true for any sign of the charge q , though we have considered q to be positive (i.e., q > 0) while deriving this equation. When q is negative (i.e., q <0), work done by the external agent in bringing test charge q0 from  to the point P is negative. This is equivalent to saying that work done by the electrostatic force in brining q0 from infinity to the point P is positive. This is perfectly in order because when q is negative, the force on the test charge q0 (which is taken as positive), is attractive. As such, the electrostatic force and displacement of q0 (from  to P) are in the same direction. 13. 0 W W Edl AB BA + = =  is consistent with choice that V = 0 at  14. The variation of E and V with r is shown in figure. 15. Though V is called the potential at a point, actually it is equal to potential difference between r and  . 16. Electric potential due to a uniformly charged ring. As 2 2 ; e e e dq dq k dq dV k V k r r R x = = = +   Since the distance of each element dq from the point P is the same, 2 2 R x + is constant.