2.3 Electric Potential

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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

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ReviewBy 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.

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Electric potential Define a scalar point function, V(r), known as the electric potential (integral version) Reference point By convention, the point r = ref is taken to be a standard reference point of electric potential, V(r) where V (r = ref ) = 0 (usually r = ). SI Units of Electric Potential = Volts If V (r)= ref = 0 @ the reference point, then V(r) depends only on point r .

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Electric potential (conti.) Electric potential difference between two points a & b

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Electric potential (conti.) Thus

The fundamental theorem for gradients states that

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Electric potential (conti.) The above equation is true for any end-points a & b (and any contour from a b). Thus the two integrands must be equalKnowing V(r) enables you to calculate E (r ) !!

Now (for electrostatics):

Thus

So, for Electrostatic problems, E(r) = 0 will always be true !EM-2.3-6

Why is E(r) specified as negative gradient of the electric potential? Consider the point charge problem

In spherical-polar coordinates

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Why is E(r) specified as negative gradient of the electric potential? (conti.)

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Why is E(r) specified as negative gradient of the electric potential? (conti.) V(r) for a point charge Q Q=+e Q=-e

Radial outward Lines of E(r)

Radial inward Lines of E(r)

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-2.3-9

Why is E(r) specified as negative gradient of the electric potential? (conti.)

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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 potential These imaginary surfaces of constant potential are known as equipotential surfaces E +q E E E V1 V2 EEM-2.3-11

E E

The equipotentials of constant V(r) are everywhere perpendicular to lines of E(r) !

Equipotentials: Arbitrary charge distribution

Consider a charged metal

Charged metal

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Electrostatic Potential and Superposition Principle We have seen that, for any arbitrary electrostatic charge distributions:

Since or

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Electrostatic Potential and Superposition Principle (conti.) Integrate from a common reference point, a = ref

Since Therefore

Note that this is a scalar sum, not a vector sum!

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Example 2.7 A uniformly charged spherical (conducting) shell of radius, R, find the electric field.

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Example 2.7 (conti.) Calculate V(r) from

use law of cosines

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Example 2.7 (conti.)

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Example 2.7 (conti.) Note that

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Example 2.7 (conti.)

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Example 2.7 (conti.) Then electric field

Thus

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POISSONS EQUATION & LAPLACES EQUATION

Poissons equation

Laplaces equation IfEM-2.3-21

Cartesian Coordinates

Cylindrical Coordinates

Spherical Coordinates

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Typical electrostatic problem Given charge distribution

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Typical electrostatic problem Given V(r)

Given E(r)

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Typical electrostatic problem : Summary

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Let h 0

BOUNDARY CONDITIONS

Example 2.4EM-2.3-26

E is discontinuous across a charged interface

Therefore

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Tangential components of E across a charged surface Let h 0

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Normal derivative of the potential V Since

But

Thus

Since

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V across a charged surface

Let h 0

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