The gradient of a scalar field, the Physical significance of the gradient, and numerical problems on the gradient of a scalar field for B.Sc Physics - Mechanics - first year first -semester

1. Vector Analysis Lecture - 2
B.Sc Physics - First Year - First Semester
Topic: Gradient of Scalar Field and
Numericals on gradient

2. ∇
Gradient of Scalar field
operated on Scalar field(Φ)
Vector field
(∇Φ)
Operating del operator on a scalar field results in gradient and the
gradient of a scalar field gives the maximum rate of change of the
field at a point

3. Φ
Level Surface
Normal direction
n̂


Level Surface (Equipotential surface):
A surface on which the value of the field
is constant (refers to a region in space
where every point in it is at the
same value.)
n
n
Grad ˆ









4. Gradient of a scalar field:
Let two level surfaces (equipotential
surfaces) S1 and S2 very close to each other.
The level surfaces are specified by
scalar values S and S + dS.
Let r and r + dr be the radius vectors
of points A and B with respect to origin O
respectively.

5. The vector drawn from A to B is
The least distance between S1 and S2 is AC.
The direction of AC is along
is unit normal vector
the length of AC is dn
The rate of change of S at A
in the direction AC =
r
d
r
OB
r
OA


 ,
r
d
r
r
d
r
OA
OB
AB 





n̂
n̂
n
S


r
d

6. But, gives the maximum rate of change at A along the
direction of AC. This quantity is called gradient.
∴
Therefore, the gradient of a scalar quantity is a vector
)
3
(
ˆ 





 n
n
S
Grads
 
dn
drCos
dn
n
S
drCos
Cos
r
d
n
r
d
n
drCos
n
S
r
d
n
n
S
r
d
gradS




























 ˆ
ˆ
ˆ
n
n
S
ˆ


∴ gradS.dr = ds -----------(4)

7. But, in cartesian coordinate system the scalar function S is taken as
S = S(x,y,z)
 
)
6
(
ˆ
ˆ
ˆ
S
gradS
eqn.(4)
and
eqn.(5)
equating
)
5
(
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ

























































z
S
k
y
S
j
x
S
i
r
d
S
dS
dz
k
dy
j
dx
i
z
S
k
y
S
j
x
S
i
ds
dz
z
S
dy
y
S
dx
x
S
ds



8. S
z
k
y
j
x
i
S
gradS 


















 ˆ
ˆ
ˆ


9. Physical Significance of Gradient
 By operating Del operator on scalar field we can get vector field.
Gradient of a scalar field gives the maximum rate of change of field
at a point(gradient of a scalar point function represents normal
vector to the level surface.)
Vector fields obtained by gradient of scalar fields are called as
Lamellar fields
eg: Electric field intensity E= - ∇V
 Gradient of gravitational potential gives gravitational field
intensity
In Lamellar fields, the line integral over a closed curve is always
zero
 Gradient is independent of coordinate system, so gradient is
invariant

10. Problem1.If the potential of a scalar field is
. Find the gradient of the field.
2z
3y
7x
φ 


Solution:
grad φ = ?
grad φ = 𝛁 φ = 𝛁 ( )
2z
3y
7x
φ 


2z
3y
7x 

 
2z
3y
7x
z
k̂
y
ĵ
x
î 
















     
z
y
x
x
k
z
y
x
y
j
z
y
x
x
i 2
3
7
ˆ
2
3
7
ˆ
2
3
7
ˆ 














k
j
i ˆ
2
ˆ
3
ˆ
7 



11. Problem 2

15. Assignment:
1.Define gradient of a scalar field and write its physical
significance
2.Derive an expression for Gradient of a scalar field
3.What are Lamellar fields, give examples for lamellar
fields.

16. Numericals :

17. Thank you