For revision points of the chapter
The charge on a proton is
e = 1.60218*10^-19 C
Coulomb's formula for the electric force between two charges.
(1) F = k*q1*q2/r²
r = separation between charges
k = constant
In SI units k is measured to be 8.98755*10^9 N-m²/C²
The constant k is often written as 1/4πε0.
The constant ε0 is called the permittivity of the space and its value is
ε0 = 8.85419*10^-12 C²/N-m²
The intensity of field is defined as Vector E = Vector F/q
(3) E = F/q
Electric field due to a point charge (Q)
(4) E = F/q = Q/4πε0r²
(5) U(r2) - U(r1) = -W = [q1*q2/4πε0]* [(1/r2) - (1/r1)]
Choosing potential energy at infinite separation as zero
(6) U(r) = U(r) - U(∞) = [q1*q2/4πε0]* (1/r)
(7) VB - VA = (UB - UA)/q
We can define absolute electric potential at any point by choosing a reference point P and saying that the potential at this point is zero.
(8) VA = VA - VP = (UA - UP)/q
The potential due to a point charge Q (placed at A) at a point P with distance AP = r is
(9) VP = Q/4πε0r
The electric potential at a point due to more than one charge in the system is obtained by finding potential due to individual charges and then adding them.
10. Relation between electric field and potential
dV = -E.dr
Integrating between limits r1 and r2
V2 – V1 = - ∫ E.dr
If we choose r1 as infinity (reference point)
V(r) = - ∫ E.dr (from infinity to r)
Hence we can calculate potential V if we know E and r.
If we know V we can find E through the relation
Ex = - ∂V/∂x
Ey = -∂V/∂y
Ez = -∂V/∂z
We can find the x,y and z components of E.
E can be written as Ex i +Ey j + Ez k
We can write dV as dV = -E dr cos θ where θ is the angle between the field E and the small displacement dr.
-dV/dr = E cos θ
15. Electric Dipole
Electric dipole moment
It is defined as a vector p = q*distance vector d
where distance vector is the vector joining the negative charge to the positive charge.
The line along the direction of the dipole moment is called the axis of the dipole.
16. Electric potential due to a dipole at a point P
Point is at distance r from the centre of the diploe (d/2) and theline joining the point P to the centre of the dipole make an angle θ with the direction of dipole movement (from –q to +q)
Potential at P due to charge –q = - [1/4πε0][q/(r + (dcos θ)/2)]
Potential at P due to charge q = [1/4πε0][q/(r - (dcos θ)/2)]
Net potential due to q and –q = [1/4πε0](qd cos θ)/r²
17. Electric field due to a dipole
Er = [1/4πε0](2p cos θ)/r³
Eθ = [1/4πε0](p sin θ)/r³
Resultant electric field at P = E= √ (Er²+ Eθ²)
= [1/4πε0](p/r³)√(3 cos²θ + 1)
19. Torque on an electric dipole placed in an electric field.
If the dipole axis makes an angle θ with the electric field magnitude of the torque = | Γ| = pE sin θ
In vector notation Γ = p × E
20. Potential energy of a dipole placed in a uniform electric field
dipole axis makes an angle θ with the electric field magnitude of the torque
Change in potential energy = U(θ) – U(90°) = -pE cos θ = -p.E
If we choose the potential energy of the dipole to be zero when θ = 90° , above equation becomes
U(θ) = -pE cos θ = -p.E