EM-2.3-1
ReviewThe electric field E(r) is a very special type of vector field
For electrostatics, the CURL of E(r) = zero, i.e.
The physical meaning of the curl of a vector field:For an arbitrary vector field A(r) , if ∇×
A(r)≠0 for all
points in space, then the vector A(r) rotates, or shears in some manner in that region of space
Curl of Whirlpool Field, ∇ ×
v (r) ≠ 0
Curl of shear Field∇ ×
v (r) ≠ 0
EM-2.3-2
Review
By use of Stokes’ Theorem
There are two implications (assuming E(r) ≠ 0 everywhere):1. everywhere (for arbitrary closed surface S).2. implies path independence of this (arbitrary)
closed contour, C.
EM-2.3-3
Electric potential
Define a scalar point function, V(r), known as the electric potential
(integral version)
Reference pointBy convention, the point r = Οref is taken to be a standard reference point of electric potential, V(r) where V (r = Οref ) = 0 (usually r = ∞).
SI Units of Electric Potential = Volts
If V (r)= Οref = 0 @ the reference point, then V(r) depends only on point r .
EM-2.3-4
Electric potential (conti.)
Electric potential difference between two points a & b
EM-2.3-5
Electric potential (conti.)
Thus
The fundamental theorem for gradients states that
EM-2.3-6
Electric potential (conti.)
The above equation is true for any end-points a & b (and any contour from a → b). Thus the two integrands mustbe equal
Now (for electrostatics):
Thus
So, for Electrostatic problems, ∇×
E(r) = 0 will always be true !
Knowing V(r) enables you to calculate E (r ) !!
EM-2.3-7
Why is E(r) specified as negative gradient of the electric potential?
Consider the point charge problem
In spherical-polar coordinates
EM-2.3-8
Why is E(r) specified as negative gradient of the electric potential? (conti.)
EM-2.3-9
Why is E(r) specified as negative gradient of the electric potential? (conti.)
Q=+e Q=-e
Radial outward Lines of E(r) Radial inward Lines of E(r)
V(r) for a point charge Q
By defining E(r) as the negative gradient, this simultaneously defines that lines of E point outward from (+) charge, and point inward for (-) charge.
EM-2.3-10
Why is E(r) specified as negative gradient of the electric potential? (conti.)
EM-2.3-11
Equipotentials: point charge
For a point charge, q, there exist “imaginary” surfaces –concentric spheres of varying radii r = R1 < R2 < R3 < …whose spherical surfaces are surfaces of constant potentialThese “imaginary” surfaces of constant potential are known as equipotential surfaces
E
E
E
E
E
EE
V1
V2
The equipotentials of constant V(r) are everywhere perpendicular to lines of E(r) !
+q
EM-2.3-12
Equipotentials: Arbitrary charge distribution
Charged metal
Consider a charged metal
EM-2.3-13
Electrostatic Potential and Superposition Principle
We have seen that, for any arbitrary electrostatic charge distributions:
Since
or
EM-2.3-14
Electrostatic Potential and Superposition Principle (conti.)
Integrate from a common reference point, a = Οref
Since
Therefore
Note that this is a scalar sum, not a vector sum!
EM-2.3-15
Example 2.7
A uniformly charged spherical (conducting) shell of radius, R, find the electric field.
EM-2.3-16
Example 2.7 (conti.)
Calculate V(r) from
use law of cosines
EM-2.3-17
Example 2.7 (conti.)
EM-2.3-18
Example 2.7 (conti.)
Note that
EM-2.3-19
Example 2.7 (conti.)
EM-2.3-20
Example 2.7 (conti.)
Then electric field
Thus
EM-2.3-21
POISSON’S EQUATION & LAPLACE’S EQUATION
Poisson’s equation
Laplace’s equation
If
EM-2.3-22
Cartesian Coordinates
Cylindrical Coordinates
Spherical Coordinates
EM-2.3-23
Typical electrostatic problem
Given charge distribution
EM-2.3-24
Typical electrostatic problem
Given V(r)
Given E(r)
EM-2.3-25
Typical electrostatic problem : Summary
EM-2.3-26
BOUNDARY CONDITIONSLet h → 0
Example 2.4
EM-2.3-27
E is discontinuous across a charged interface
Therefore
EM-2.3-28
Tangential components of E across a charged surface
Let h → 0
EM-2.3-29
Normal derivative of the potential V
Since
But Thus
Since
EM-2.3-30
V across a charged surface
Let h → 0
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