mathmatical physics

1. NPTEL – Physics – Mathematical Physics - 1
Lecture 2
Physical examples of gradient
1. An electric dipole moment 𝑝⃗ located at origin, creates a potential at 𝑟⃗ given
by,
𝑝.𝑟⃗
V(𝑟⃗) = K , where K is a constant.
𝑟3
Find the electric field 𝐸⃗⃗ = - ∇⃗⃗ V (𝑟⃗). Where ∇⃗⃗ is an operator given by ⃗∇⃗= ( iˆ 𝑑
+ ˆj 𝑑
+ kˆ 𝑑
)
𝑑𝑥 𝑑𝑦 𝑑𝑧
Solution
𝐸⃗⃗ = −𝑝 (𝑥̂
𝑑
+ 𝑦̂
𝑑
+
𝑑
)
𝑑𝑥 𝑑𝑦 𝑑𝑧
(𝑥 + 𝑦 + 𝑧)
(𝑥2 + 𝑦2 + 𝑧2) ⁄2
3
The 𝑥̂ component yields,
= −𝑝 [
1
(𝑥2 + 𝑦2 + 𝑧2) ⁄2 (𝑥2 + 𝑦2 + 𝑧2) ⁄2
3 5
−
3𝑥2
] x̂
Putting the contribution from ŷ and ẑ
𝐸⃗⃗ =
3𝑝
(𝑥2 xˆ + 𝑦2 yˆ + 𝑧2 zˆ )
−
𝑝 rˆ
𝑟5 𝑟3
2. For a position vector 𝑟⃗, calculate ∇⃗⃗𝑟𝑛 and use this result to calculate
⃗∇⃗( ).
1
𝑟
Solution
∇⃗⃗𝑟𝑛 = ( 𝑥̂ 𝑑
+
yˆ 𝑑𝑥
𝑑
𝑑𝑦 𝑑𝑧
+ ẑ
𝑑
)𝑟𝑛
= 𝑥̂ 𝑛𝑟𝑛−1 𝑑𝑟
+ yˆ 𝑛𝑟𝑛−1 𝑑𝑟
+ 𝑧̂ 𝑛𝑟𝑛−1 𝑑𝑟
𝑑𝑥 𝑑𝑦 𝑑𝑧
= 𝑛𝑟𝑛−1[ xˆ 𝑑𝑟
+
yˆ 𝑑𝑥
𝑑𝑟
+ ẑ 𝑑𝑟
]
𝑑𝑦 𝑑𝑧
Using 𝑟2 = 𝑥2 + 𝑦2 + 𝑧2
𝑑𝑟 𝑥 𝑑𝑟
𝑑𝑥 𝑟 𝑑𝑦 𝑟 𝑑𝑧 𝑟
= , = , =
𝑦 𝑑𝑟 𝑧
∇⃗⃗𝑟𝑛 = 𝑛𝑟𝑛−2 (𝑥 xˆ + 𝑦 yˆ + 𝑧
zˆ )
= 𝑛𝑟𝑛−2𝑟⃗
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2. NPTEL – Physics – Mathematical Physics - 1
To calculate ∇ ( ), put n =-1
⃗
⃗
1
𝑟
∇⃗⃗
(
1 𝑟
) =
𝑟⃗
𝑟3
The potential due to a point change goes as , so electric field goes as
1
𝑟
𝑟⃗
𝑟3
Exercise
Two identical point charges of magnitude Q at (1, 0, 0) and (-1, 0, 0). The potential in the xy
plane is given by,
V(x, y, z =0) = [ +
𝑄 1 1
4𝜋𝜖0 √(𝑥−1)2+𝑦2 √(𝑥+1)2+𝑦2 ]
Find the electric field using, 𝐸⃗⃗ = - 𝛻⃗⃗𝑉.
Formal definition and properties of a Gradient
Thus gradient represents a direction that is perpendicular to the surface 𝑔(𝑟⃗). ⃗∇⃗𝑔(𝑟⃗) = 0
denotes a minimum or a maximum or a saddle point (saddle point is defined as the one that is minimum in
one direction, but is maximum in the other direction.
Let us consider a scalar function 𝜑(𝑥, 𝑦, 𝑧). A change in 𝜑 is given by,
𝑑𝜑 = (𝜕𝜑
) 𝑑𝑥 + (𝜕𝜑
) 𝑑𝑦 + (𝜕𝜑
) 𝑑𝑧 ∗
𝜕𝑦 𝜕y 𝜕z
Example
Find ∇⃗⃗ 𝜑 and |∇⃗⃗ 𝜑| for 𝜑 = 2𝑥𝑧4- 𝑥2y at the point (2,-2,-1). Solution
𝜑 = 2𝑥𝑧4 - 𝑥2𝑦
∇⃗⃗𝜑 = xˆ [2𝑧4 − 2𝑥𝑦] + yˆ [−𝑥2] + zˆ [8𝑥𝑧3]
|⃗⃗∇⃗⃗ 𝜑|(2,−2,−1) = 10 xˆ −4𝑦 − 16𝑧3
Then |∇⃗⃗𝜑| = 2√93
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3. NPTEL – Physics – Mathematical Physics - 1
Exercise problem
The above equation can be written as,
𝑑𝜑 = ∇⃗ 𝜑. 𝑑𝑙 with ∇⃗ 𝜑 = xˆ 𝜕𝜑
+ yˆ 𝜕𝜑
+ zˆ 𝜕𝜑
and 𝑑𝑙 is a differential length
element
𝜕𝑥 𝜕𝑦 𝜕𝑧
given by,
𝑑𝑙 = 𝑥̂ 𝑑𝑥 + 𝑦̂ 𝑑𝑦 + 𝑧̂ 𝑑𝑧
The ∇⃗ 𝜑 is a vector where magnitude is the maximum value of 𝑑𝜑
and whose direction is
the 𝑑𝑙
direction in which is maximum.
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𝑑𝜑
𝑑𝑙
Exercise
Find a unit vector perpendicular to the surface 𝑥2 + 𝑦2 + 𝑧2 = 1 at the point (4,2,3).
Before we wind up our discussion on gradient, we wish to remind the reader that gradient of a function 𝜑,
that is ∇⃗⃗𝜑 is perpendicular to the surface 𝜑 = constant. Thus, considering a physical situation that four
sound boxes are located at four corners of the room, each emitting a music of different frequency and
intensity, if we ask a question that what is the distribution of sound intensity in the room. Gradient of this
intensity function will tell you the direction along which the change is most rapid.