Electromagnetism for Materials (MScE2108)
Targeted group 2nd year
Semester II
Year: 2022
Chapter 4: scalar potential
Reference : Basic Electromagnetism and Materials, Categories,2nd edition, 2006, Springer,ISBN 10: 0387302840
ISBN 13:9780387302843
Out line of the chapter
∞ Definition and properties of Scalar potential
∞ Uniform spherical charge distribution
∞ Uniform Line Charge distribution
∞ Scalar potential and Energy
The Scalar Potential
 We described the electrostatic effect by using the vector
field E. however, it is also substantially possible to obtain
the same information in terms of Scalar fields
 Recall that E can be expressed by:
𝑵
෡𝒊
𝒒𝒊′ 𝑹
𝑬(𝒓) = ෍
𝟒𝝅𝜺𝒐𝑹𝟐𝒊
𝒊=𝟏
෡
𝑹
1
𝒊
𝟐 −𝛻
𝑹𝒊
𝑅𝑖
 𝜙 Is represented a scalar potential/ electrostatic potential,
it’s unit is volt.
 For E Volt/meter, Newton/Coulomb; combining the two
1Volt=1Joul/Coulomb
Stokes’ Theorem
Example
1. A nonuniform electric field is given by E = xax + yay + 2zaz V/m. A charge of 3C is
transferred from point A(1, 2, 3) to point B(2, 4, 5) along the differential length of the line.
Consider the differential length in Cartesian coordinates and determine the potential.
2. A nonuniform electric field is given by E = yax +xay +2yaz V/m. A charge of 2C is
transferred from point A(1, 0, 3) to point B(2, 1, 3) along the straight line from point A to
point B. Calculate the potential.
 If the source charge have continuous distribution:
 To avoid ambiguity better to define the scalar potential :
 Other way of expressing the relation between E and 𝜙:
𝟐
𝜟𝝓 = − න 𝑬. 𝒅𝒔
𝟏
 From both eq.(5.5) and (5.11) we can see that E is a
conservative fields
 Equipotential surfaces are a surface on which the potential
fields, 𝝓, are constant
 Direction of E is perpendicular to
Equipotential surfaces, 𝝓.
 Density of lines are proportional to the
magnitude of E.
 The magnitude of E is graters at the region
were they are close each other than the region
where the lines are farther apart
 Let consider a point charge Q located at the r’
 The equipotential surface will obtained by:
 The surfaces are a spherical
with center Q and,
 The Electric fields are
perpendicular to the
equipotential surfaces with
the direction of decreasing 𝜙
 𝝓3> 𝝓2> 𝝓1
 Other approach obtained combining the followings
&
 A scalar potential satisfies such differentials equation said
to be Poisson's Equation
 If the volume charge density is zero in the given region the
equation (5.15) resembles Laplace’ equation
Uniform Spherical Charge Distribution
 Recall that the total charge Q on the sphere of radius ‘ɑ’
with constant charge density ρ
𝟑
𝝆=
𝟑Q
𝟒𝝅ɑ
 The charge is uniformly distributed over a sphere and
hence the volume charge density ρ is constant, i.e.,
independent of variable r.
𝑹 = 𝒛ො𝒛 − 𝒓′ො𝒓′
𝑹 ⋅ 𝑹 = (𝒛ො𝒛 − 𝒓′ො𝒓′ ) ⋅ (𝒛ො𝒛 − 𝒓′ො𝒓′ )
𝑹𝟐 = 𝒛𝟐 + 𝒓′ 𝟐 − 𝟐𝒛𝒓′(ො𝒛 ⋅ 𝒓ො ′ )
𝒛ො ⋅ 𝒓ො ′ =Cos𝛉
𝑹 = (𝒛𝟐 + 𝒓′ 𝟐 − 𝟐𝒛𝒓′Cos𝛉)1/2
 Two cases should be considered; out side and inside of the
sphere
 If z > ɑ then (z-r’) > 0
=2/z
 If z < ɑ then (z-r’) < 0
=2/r’
Then the inside potential is
The potential inside and outside the sphere
To find electric field inside and outside the sphere is by inserting
the potential in to negative gradient
The scaler potential and energy
The potential is related to changes in energy
Under the equilibrium
conditions
The work done by the external agent responsible for
mechanical force
The work done is equal to the charge times changes in potentials
So potential energy is
The SI unit is joule