Gauss's Law for Electric Fields Flashcards
The Fundamental Forces
- gravitational forces
- electromagnetic forces
- strong nuclear force
- weak nuclear force
Field
- interaction at a distance
- certain objects set up a field around them which other particles interact with
- they are illustrated with arrows and field lines
Fields and Flow
-nothing is flowing in a magnetic or electric field, but the mathematical relationships between various quantities in these fields will be the same as in fluid flow
Electrostatics and Magnetostatics
- stationary charges have constant electric fields; electrostatics
- steady currents have constant magnetic fields; magnetostatics
Steady Currents
- a steady current I, means a continuous flow that has been going on forever without charge piling up anywhere
- a moving point charge cannot constitute a steady current since it is here one instant and gone the next
Coulomb’s Law
Electric Field of a Point Charge
|E(|r) = kq/r²r^
k = 1/4πε0 = 8.99x10^9 Nm²/C²
Electromagnetism - Superposition Principle
-the net response at a given place is the sum of the responses which would have been caused by each stimulus individually
Continuous Charge Distribution
Charge Density
ρ = Q/V dq = ρdV
Continuous Charge Distribution
Electric FIeld
|E(|r) = ∫ kdq/r² r^ = ∫ kρ/r² r^ dV
The electric field is the integral over all charge of kdq/r² or the integral over the entire volume of kρ/r².
Gauss’s Law for Electric Fields - Integral Form
∮(|E . n^)dA = Q/ε0
The integral of electric field in the direction perpendicular to the surface across a closed surface is equal to the total charge enclosed by that surface divided by the permitivity of free space
OR
An electric charge produced an electric field and the flux of that field passing through any closed surface is proportional to the total charge contained within that surface.
Solid Angle
ΔA’ = r² x ΔΩ
where A’ is the are on the surface of the sphere
r is the radius of the sphere and Ω is the solid angle
- to go right around a sphere or any closed surface you have to travel around by Ω=4π steradians
- this is why the surface area of a sphere is 4πr²
Gauss’s Law for Electric Fields - Integral Form
Multiple Point Charges
∮(|E . n^)dA = ∮ Σ|E.n^ dA = ∮|E1.n^dA + ∮|E2.n^dA + …
=q1/ε0 + q2/ε0 + …
=Q/ε0
Gauss’s Law for Electric Fields - Integral Form
Charge Outside of the Closed Surface
-the net flux from any charge outside of the closed surface is zero as the flux into the surface from that charge is equal to the flux out, so net flux is zero.
Gauss’s Law for Electric Fields
Sources and Sinks
-net flux will tell you whether inside a closed surface you have a net source or sink of electric field lines and whether the net flow is out or in
Is a positive charge a source or a sink?
-positive charge acts as a source of field lines like a tap within the enclosed volume generating the flow out of the surface containing that volume
Is a negative charge a source or a sink?
-negative charge acts as a drain of field lines, like a sink within the enclosed volume generating a flow into the surface containing that volume
Electric Fields - Definition of Divergence
div(|E) = ∇ . |E =
V->0 lim 1/ΔV ∮ |E . n^dA
-this limit gives the next flux per unit volume at a point
|∇ . |E = ∂Ex/∂x + ∂Ey/∂y + ∂Ez/∂z
Gauss’s Law for Electric Fields
Differential Form
|∇ . |E = ρ/ε0
Gauss’s Law for Electric Fields
Integral to Differential Form
-start with Gauss's Law: ∮(|E . n^)dA = Q/ε0 -divide both sides by ΔV 1/ΔV*∮(|E . n^)dA = Q/ΔV *1/ε0 -the left hand side is equal to ∇ . |E -right hand side is equal to ρ/ε0, where ρ is the charge density -this gives the differential form of Gauss's Law ∇ . |E = ρ/ε0
Gauss’s Law for Electric Fields
Description of Differential Form
- the electric field produced by electric charge diverges from positive charge and converges upon negative charge
- the only places at which divergence of the electric field is not zero are the locations at which charge is present
Where does divergence or convergence occur in an electric field?
- the only places where divergence or convergence occurs is at point charges
- even if at a point in the field the lines appear to be moving away from each other, if you took a small volume the next flux in and out would still be 0 so by definition, no divergence or convergence
Electric Flux Through a Closed Surface S
|E uniform and perpendicular to S
φ = |E| * (surface area)
Electric Flux Through a Closed Surface S
|E uniform and at an angle to S
φ = |E . ^n
Electric Flux Through a Closed Surface S
|E non-uniform and at variable angles to S
φ = ∫ |E . ^n dA
Rule for Drawing Electric Fields
1) electric field lines must originate on positive charge and terminate on negative charge
2) the net electric field at any point is the vector sum of all electric fields present at that point
3) electric field lines can never cross since that would indicate that the field point in two directions at the same location which violates rule 2
4) electric field lines are always perpendicular to the surface of a conductor in equilibrium
Electric Field of a Conducting Sphere
|E = 1/4πεo * Q/r² * ^r
-outside the sphere a distance r from the centre
|E = 0
-inside
Electric Field of a Uniformly Charged Insulating Sphere (radius=ro)
|E = 1/4πεo * Q/r² * ^r
-outside the sphere, a distance r from the centre
|E = 1/4πεo * Qr/(ro)³ * ^r
-inside the sphere, a distance r from the centre
Electric Field of an Infinite Line Charge
linear charge density=λ
|E = 1/2πεo * λ/r * ^r
-a distance r from the line
Electric Field of an Infinite Flat Plane
surface charge density = σ
|E = σ/2εo ^n
Uses of the Integral Form of Gauss’s Law for Electric Fields
1) given information about a distribution of electric charge, you can find the electric flux through a surface enclosing that charge
2) given information about the electric flux through an enclosed surface you can find the total electric charge enclosed by that surface
Insulator/Dielectric
Definition
-in a dielectric material, charges do not move freely but can be displaced from their equilibrium position
Special Gaussian Surface
- in certain cases a surface can be constructed such that it is possible to calculate the electric field
- such a surface must meet two conditions:
1) the electric field must be either parallel or perpendicular to the surface normal at all points on the surface
2) the electric field must be constant or zero over the surface
Describe the fundamental difference between the integral form and the differential form of Gauss’s law for electric fields
-the differential form deals with divergence of the electric field and charge density at INDIVIDUAL POINTS in space whereas the integral form entails the integral of the normal component of the electric field OVER A SURFACE
Uses of the Differential Form of Gauss’s Law for Electric Fields
1) if spatial variation of the vector electric field is known at a specified location, then you can find the charge density at that location
2) if volume charge density is known, the divergence of the electric field may be determined
