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Unit 1

The document discusses various concepts in vector calculus, including scalar and vector fields, and operations such as gradient, divergence, and curl. It explains the Gauss divergence theorem, Stokes' theorem, and Gauss's law in electrostatics, detailing their mathematical formulations and implications. Additionally, it covers the Poisson and Laplace equations, highlighting their relationships in the context of electric fields and potentials.

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ashmitbhardwaj06
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9 views11 pages

Unit 1

The document discusses various concepts in vector calculus, including scalar and vector fields, and operations such as gradient, divergence, and curl. It explains the Gauss divergence theorem, Stokes' theorem, and Gauss's law in electrostatics, detailing their mathematical formulations and implications. Additionally, it covers the Poisson and Laplace equations, highlighting their relationships in the context of electric fields and potentials.

Uploaded by

ashmitbhardwaj06
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Field:

Some region in space where some quantity is specified

- Scalar field: each point in field is scalar quantity


- Vector field: each point in field is vector quantity
Gradient of a scalar field
Gives the direction in which function increases at the
maximum rate.

Field: Scalar field


Output: Vector quantity
Divergence
Divergence
How much field / flux spreads out or diverge from a given point

Field - Vector
Output - Scalar

Point will act Point will act Field is


as source as sink solenoidal
Curl
Curl at a point in a field is a measure of the rotation / circulation of
vector field
Field - Vector
Output - Vector
Curl

Conservative field is the field in which work done between two different
points is independent of path followed or work done in a closed path is
zero. Example - Electrostatic field (electric field independent of time)
Curl
Find Curl ( F )⃗ F ⃗ = xy 2 i ̂ + yz 2 j ̂ + zx 2k̂

Curl ( F )⃗ =
Gauss divergence theorem
The flux of a vector field F ,⃗ over any closed surface S, is equal to the volume
integral of the divergence of that vector field over the volume V enclosed by
the surface S.

Surface should be closed

Connects surface integral to volume integral


Stoke’s theorem

The surface integral of the curl of a vector field F ,⃗ over any surface S, is equal
to the line integral of F ⃗ around the closed curve form the periphery of the
surface S.

curve should be closed and surface should be open

Connects line integral to surface integral


Gauss law (Electrostatic)

It states that the electric flux through any closed


surface is proportional to the total electric charge
enclosed by this surface.

Integral form of Gauss Law


Gauss law (differential
Flux form)
According to Gauss law

⃗ ⃗ 1
∫∫∫ ∫ ∫ ∫
Where:
E - Electric field ( ∇ . E )dv = ρdv
DA - surface element
Q - Charge enclosed in closed surface
ϵ0
ε₀ - Permittivity of free space

∫∫∫
Qenclosed = ρdv Where ρ is volume charge density
Two integral will be equals for any volume only if integrands will be equal

E ⃗ . DA ⃗ =
1
∫ ∫ ∫ ∫
ϕ= ρdv
ϵ0
s

∇⃗ . E ⃗ =
By using Gauss divergence theorem ρ

E ⃗ . DA ⃗ =
ϵ0
⃗ ⃗ 1
∫ ∫∫∫ ∫ ∫ ∫
ϕ= ( ∇ . E )dv = ρdv
ϵ0 Differential form of Gauss Law
s
Poisson & Laplace equations
Where:

∇⃗ . E ⃗ =
Differential form of Gauss Law
ρ E - Electric field
DA - surface element
Q - Charge enclosed in closed surface

ϵ0 ε₀ - Permittivity of free space


ρ volume charge density
V electric potential
∇² is Laplacian operator

E ⃗ = − ∇ ⃗V

∇ ⃗ . ( − ∇ ⃗ V) =
ρ
In absence of charges,
ϵ0 Poisson equation will reduce to Laplace Equation
ρ=0
2 ρ
∇ V=−
ϵ0 ∇2 V = 0
Poisson Equation Laplace Equation

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