1. Line integral of Electric field: Electric Potential
Electric field at a field point r, due to a point charge at origin:
After the discussion of area integral of E over the surface (Flux q/ε0 ), what about Line integral of E from some point a to some other point b?
And if a=∞, b=r
The electric potential at a distance r from the point charge is the work done per unit charge in bringing a test charge from infinity to that point.
The integral around a closed path is zero
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2. Relation between E and V
As fundamental theorem for gradient is
Surface over which Potential is constant is called an equi-potential surface.
Reference point : convention at infinity.
Superposition principle: V=V1+V2+…..
Unit: Nm/C or Joule/C or VOLT
Curl of E ?
The integral around a closed path is zero
Using Stokes’ theorem =>
Hence curl of E
In electrostatics only means no moving charge or current
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3. Electric Potential due to a Point Charge
Electric Field:
Electric Potential:
Convention: V=0 at infinite r
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4. Electric Potential due to a Point Charge
Electric potential at a
distance r from a positive charge Q
Electric potential at a
distance r from a negative charge Q
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5. Electric Potential due to a System of Point Charges
For a system of point charges Qi at distances ri from a point P: Q1 Q4 Q3 Q2 P r1 r2 r3 r4
… an algebraic sum of scalars!
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6. Electric Potential due to a Continuous Charge Distribution
If the charges are not discrete but are continuously distributed over some object or region, the summation is replaced by an integral:
where r is the distance from a point P where the potential is to be determined to an element of charge dq and the integral is taken over the entire distribution of charge.
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7. Example of Continuous Charge Distribution: Ring of Charge
Calculate the electric potential V at a distance x along the axis of a thin, uniformly charged ring of radius R carrying a total charge Q.
Uniformly Charged Ring
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8. Example of Continuous Charge Distribution: Uniformly Charged Disk
Calculate the potential on the axis at a distance x from a uniformly charged disk of radius R.
Divide the disk into thin rings of radius r and thickness dr.
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9. Determination of the Electric Field from the Electric Potential
If the electric potential is known in space, the electric field may be determined from it.
Relationship between electric potential and electric field:
The component of E in any direction is the negative of the rate of change of the electric potential with the distance in that direction.
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10. Potential for a Spherical Shell of Charge Q
V must be continuous, so inside the shell (r < R),
On surface of sphere (r = R),
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11. Potential for a Spherical Shell of Charge
Outside:
Inside:
Inside the shell, there is no electric field, so it takes no work to move the test charge around inside the shell, therefore the electric potential is the same everywhere inside the shell. V=kQ/R
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12. Electric Potential and Electric Field of a Hollow Conducting Sphere of Radius R and Net Charge Q
How would the electric potential and electric field compare for a solid conducting sphere of the same radius with the same net charge?
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13. Electric field at r > R:
Find the potential inside and outside a uniformly charged solid sphere of radius R and total charge q. Use infinity as your reference point. Sketch V(r) .
Electric field at r > R:
Electric potential at r > R: R
Electric field at r < R:
Electric
potential
at r < R:
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14. Equipotential Surfaces and Lines
The electric potential is a scalar characteristic of the electric field (a vector quantity).
Regions of space at the same electric potential are called equipotential surfaces.
Point Charge
Uniform Electric Field
Equipotential (blue dashed) lines indicate where the equipotential surfaces intersect the page and are always perpendicular to the electric field E.
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15. Equipotential surfaces are always perpendicular to electric field lines.
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16. Electric Potential Energy
The change ΔU in the potential energy is given by
ΔU = - Wc = - ∫Fcds
Relating this to a force exerted by an electric field E on a point charge qo is
Fe = qo E
We define the change ΔU in the electric potential energy as
ΔU = - ∫ qoE ds
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17. - Wc = ΔU = - ∫ qoE ds
The electric potential energy depends upon the charge placed in the electric field. To quantify the potential energy in terms of only the field itself it is more useful to define it per unit charge
-Wc/qo =
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18. Poisson's and Laplace's Equation
The electric field is related to the charge density by the divergence relationship
and the electric field is related to the electric potential by a gradient relationship
Therefore the potential is related to the charge density by Poisson's equation
where DEL SQUARE is called Laplacian operator.
In a charge-free region of space, this becomes Laplace's equation
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19. Conversion from one to another
Summary:
Conversion from one to another ρ E V
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20. Potential Difference (voltage)
in terms of E: and
Spherical symmetry: V = V (r)
Potential energy: U = q V (joules)
Equi-potential surfaces: perpendicular to field lines
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21. Electric Dipole finite dipole as
p = qs is the dipole moment for a finite dipole +q -q r+ r r- q s
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22. Multipole expansion
Legendre polynomials: coefficients of expansion in powers of
where and
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23. Multipole expansion of V: using
with
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24. permittivity material
Linear Dielectrics
D, E, and P are proportional to each other for linear materials
e 0 -
permittivity space e
permittivity material k
e / e 0
dielectric constant ce
k - 1
susceptibility
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25. Capacitors (vacuum) Capacitor: charge +Q and –Q on each plate
define capacitance C : Q = C DV
with units 1F = 1C /1V E d
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26. Energy in dielectrics vacuum energy density: dielectric:
Energy stored in dielectric system:
energy stored in capacitor:
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27. charging a capacitor C: move dq at a given time from one plate to the other adding to the q already accumulated
in terms of the field E
 energy density
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28. Definition of Capacitance
Capacitors
1. Capacitors are devices that store electric charge
2. The capacitor is the first example of a circuit element
A circuit generally consists of a number of electrical components (called circuit elements) connected together by conducting wires forming one or more closed loops
Definition of Capacitance
The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors
C is always positive
 The SI unit of capacitance is a farad (F)
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29. Makeup of a Capacitor -Q +Q
A capacitor is basically two parallel conducting plates with air or insulating material in between.
A potential difference exists between the conductors due to the charge
The capacitor stores charge E V0 d V1
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30. More About Capacitance
 Capacitance will always be a positive quantity
 The capacitance of a given capacitor is constant
 The capacitance is a measure of the capacitor’s ability to store charge
 The Farad is a large unit, typically you will see microfarads (mF) and picofarads (pF)
 The capacitance of a device depends on the geometric arrangement of the conductors
Metallized Polyester Film Capacitors
Ceramic Capacitors
Polystyrene Film Capacitors
Electrolytic Capacitors
Electric Double Layer Capacitors (Super Capacitors)
Variable Capacitors
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31. Capacitance – Isolated Sphere
Capacitance – Parallel Plates
Energy Stored in a Capacitor
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32. Capacitors in Series C1 C2 + - C3 +Q A -Q B a b V3 V2 V1 Vab Ceq +Q -Q
Substituting for V1, V2, and V3:
Substituting for V:
Dividing both sides by Q:
Equivalent capacitance
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33. Capacitors in Parallel+ - V a Q3 Q2 Q1
Capacitors in Parallel
Ceq + - V a Q
Substituting for Q1, Q2, and Q3:
Dividing both sides by V:
Equivalent capacitance
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