2.3 Electric Potential

Review „ The electric field E(r) is a very special type of vector field ¾ 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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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.

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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 = 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 .

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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 equal Knowing 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

Radial outward Lines of E(r)

„ Q=-e

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

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

E +q

E

E

V1 V2 E

E E

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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 = Οref

„ 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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POISSON’S EQUATION & LAPLACE’S EQUATION

„ Poisson’s equation

„ Laplace’s equation If EM-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 2.4 EM-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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