electrostatic potential & capacitance

electrostatic potential

• the potential at any point is the amount of work done to bring a unit positive test charge (without acceleration) from infinity to that point

• V = W/qo

• scalar

• unit → V / J/C

• dimensions → [ML²T^-3A^-1]

definition of 1 volt

1 volt is the electrostatic potential at a point when 1J of work is done to bring 1C of positive charge from infinity to that point

electrostatic potential difference

• it is the work done to bring a unit positive charge from one point to another

• \Delta V = Va - Vb (potential difference between a and b) = Wab/ qo

electron-volt

• 1 eV is equal to the work done to move a single charge e (electron/ proton) through a potential difference of 1eV

• 1eV = 1.6 × 10^-19 J

conservative nature of electrostatic force

the work done by electrostatic forces dont depend on the path along which a charge is moved

(show proof with diagram)

electrostatic potential due to a point charge

V = Kq/r

(derive this, and draw the graph of V-r variation)

NOTE: the equation shows that at equal distances from q, V is same. hence, it is spherically symmetric

electric potential due to a system of charges

it is the algebraic sum of all potentials due to individual charges at that point

V = V1 + V2… + Vn

electric potential due to a continuous charge distribution (in terms of volume charge density)

\Delta V=q/ \rho

electric potential in a conducting spherical shell

• outside at a distance r: we know that outside, the electric field is as if the entire charge is concentrated at the centre. hence,

  • V = Kq/r

• inside the shell: inside, electric field is zero. hence, the potential is constant and is equal to the value on the surface, at every point

  • V = Kq/R (where R = radius)

electric potential due to a dipole at any point

V = Kpcos\theta/ r²

vectorically,

V = Kp.r^/r²

(derive this)

electric potential due to a dipole on its axial line

V = +- Kp/r²

and, \theta = 0 (for +) or \pi (for -)

(derive this)

electric potential due to a dipole on its equatorial line

V = 0

and, \theta = \pi/2

(derive this)

difference between electric potential due to dipole and single charge

• due to dipole, it depends not just on r but also angle between position vector r and dipole moment vector p

• due to dipole, it falls off at a distance r as 1/r² and not 1/r

equipotential surfaces

• they are surfaces which have same electrostatic potential at every point

• they can be drawn through any region where theres an electric field

• if all points at the same potential in the field are joined, they form an equipotential surface

• the shape due to

  • line charge - cylindrical

  • point charge - spherical

properties of equipotential surfaces

• they never intersect, as it would give 2 directions of electric field which isnt possible

• they are closely spaced in region of strong field and widely spaced in region of weak field

• eq. surface through a point is normal to the E at that point, and is directed from one eq. surface at higher potential to the other at lower

• no work is done to move a charge on the surface

• for a field E along x-axis, the surface is parallel to yz-plane and such.

equipotential surfaces in different cases

• by a point charge/ spherically symmetric charge distribution: it is a family of concentric spheres

• for a uniform electric field: parallel plates

• due to 2 identical positive charges: just…. look at the diagram

• due to an electric dipole: same as above

(draw the diagram for each case)

relation between electric field and potential

E= - \Delta V/\Deltas = -dV/ds = -(potential gradient)

hence, we conclude,

• electric field is in the direction where potential drop is steepest

• its magnitude is equal to change in potential per unit displacement normal to the equipotential surface at the point

(derive this)

electrostatic potential energy

• it is the total amount of work done in bringing different charges to their respective positions from infinitely large separations

• unit → J

• scalar

electrostatic potential energy of a system of 2 charges

U = W = Kq1q2/ r²

• if q1q2 > 0, U is +ve, and it means both charges have same sign

  • in bringing them closer, work is done against the force of repulsion, so that the U increases, and vice versa during seperation

• if q1q2 < 0, U is -ve, and it means both charges have opp. sign

  • in bringing them closer, U decreases and in bringing them further, it decreases

electrostatic potential energy of a system of 3 charges

conductor and its properties

they are materials in which charge can move easily. they have more free electrons

• E inside them is 0

• at the surface, E is perpendicular to it at every point

• V is constant, both inside and on surface

• \sigma is different at different points

dielectrics

• they are insulators which transmit the electric effect without conducting

• under E, a dipole moment is induced on it and a net charge appears on its surface

• these charges produce an electric field Eo opp. to E

• so, the field inside dielectric is reduced

Enet = E - Eo

(draw diagram)

capacitors

• it is a system of 2 conductors separated by an insulator

• they have charges +Q and -Q with potential difference V = V1 - V2 between them

C = Q/V

• unit → farad (F)

• dimensions → [M^-1L^-2T^4A²]

• it depends on shape, size and seperation