Podcast
Questions and Answers
How does the electric flux through a closed surface change if the net charge enclosed by the surface is zero?
How does the electric flux through a closed surface change if the net charge enclosed by the surface is zero?
- The net electric flux is always negative.
- The net electric flux is equal to the surface area.
- The net electric flux is always positive.
- The net electric flux is zero. (correct)
What happens to the electric flux through a box if the charge enclosed within the box is doubled, while the size of the box remains the same?
What happens to the electric flux through a box if the charge enclosed within the box is doubled, while the size of the box remains the same?
- The electric flux is halved.
- The electric flux remains the same.
- The electric flux is doubled. (correct)
- The electric flux becomes four times greater.
Under what condition is electric flux through a surface equal to zero, even when a uniform electric field is present?
Under what condition is electric flux through a surface equal to zero, even when a uniform electric field is present?
- When the surface is face-on to the electric field.
- When the surface is tilted at a 45-degree angle to the electric field.
- When the surface is edge-on to the electric field. (correct)
- When the electric field is non-uniform.
What is the significance of recognizing symmetry when using Gauss's law to calculate electric fields?
What is the significance of recognizing symmetry when using Gauss's law to calculate electric fields?
A closed surface contains both positive and negative charges. Which of the following statements is true regarding the electric flux through the surface?
A closed surface contains both positive and negative charges. Which of the following statements is true regarding the electric flux through the surface?
Considering a uniform electric field, how does tilting a surface from a face-on orientation by an angle $\phi$ affect the electric flux through it?
Considering a uniform electric field, how does tilting a surface from a face-on orientation by an angle $\phi$ affect the electric flux through it?
A disk of radius 0.10 m is oriented with its normal at 30 degrees to a uniform electric field E with a magnitude of $2.0 \times 10^3$ N/C. What modification will result in zero electric flux through the disk?
A disk of radius 0.10 m is oriented with its normal at 30 degrees to a uniform electric field E with a magnitude of $2.0 \times 10^3$ N/C. What modification will result in zero electric flux through the disk?
If the net charge inside a box is zero, which of the following scenarios would result in zero net electric flux through the surface of the box?
If the net charge inside a box is zero, which of the following scenarios would result in zero net electric flux through the surface of the box?
What is the electric flux through a sphere of radius $r$ that encloses a charge of +3.0 nC?
What is the electric flux through a sphere of radius $r$ that encloses a charge of +3.0 nC?
How does the electric field (E) relate to electric flux ($\Phi$) in Gauss's Law?
How does the electric field (E) relate to electric flux ($\Phi$) in Gauss's Law?
What does the dot product in the integral form of Gauss's law, $\oint \vec{E} \cdot d\vec{A}$, tell you about the relationship between the electric field $\vec{E}$ and the area vector $d\vec{A}$?
What does the dot product in the integral form of Gauss's law, $\oint \vec{E} \cdot d\vec{A}$, tell you about the relationship between the electric field $\vec{E}$ and the area vector $d\vec{A}$?
If Gauss's law is expressed as $ \Phi = \oint_S \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$, what does $q_{enc}$ represent?
If Gauss's law is expressed as $ \Phi = \oint_S \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$, what does $q_{enc}$ represent?
According to Gauss's Law, what parameter affects electric flux?
According to Gauss's Law, what parameter affects electric flux?
A spherical Gaussian surface encloses a point charge +q centered within it. A second identical spherical Gaussian surface also encloses the same charge, but this time the charge is not centered. How does the electric flux through the second surface compare to that through the first?
A spherical Gaussian surface encloses a point charge +q centered within it. A second identical spherical Gaussian surface also encloses the same charge, but this time the charge is not centered. How does the electric flux through the second surface compare to that through the first?
Two point charges, +q and -q, are positioned near each other. Which closed surface(s) will have net electric flux equal to zero?
Two point charges, +q and -q, are positioned near each other. Which closed surface(s) will have net electric flux equal to zero?
Under electrostatic conditions, where does any excess charge reside on a conductor?
Under electrostatic conditions, where does any excess charge reside on a conductor?
Under electrostatic conditions, what is the electric field inside a conductor?
Under electrostatic conditions, what is the electric field inside a conductor?
How does the electric field outside a spherical conductor under electrostatic conditions compare to that of a point charge?
How does the electric field outside a spherical conductor under electrostatic conditions compare to that of a point charge?
In the context of Gauss' Law, what considerations are important when dealing with problems involving symmetry?
In the context of Gauss' Law, what considerations are important when dealing with problems involving symmetry?
An infinite positively charged wire has a charge density $\lambda$. If you construct a cylindrical Gaussian surface around the wire, what is the angle between the electric field E and the area vector dA at the end caps of the cylinder?
An infinite positively charged wire has a charge density $\lambda$. If you construct a cylindrical Gaussian surface around the wire, what is the angle between the electric field E and the area vector dA at the end caps of the cylinder?
What is the purpose of the integration over area in the context of Gauss's law?
What is the purpose of the integration over area in the context of Gauss's law?
When applying Gauss's Law to find the electric field due to an infinite line of charge, why is a cylindrical Gaussian surface typically chosen?
When applying Gauss's Law to find the electric field due to an infinite line of charge, why is a cylindrical Gaussian surface typically chosen?
In the application of Gauss's law to find the electric field around an infinite wire with uniform charge density, which statement is true regarding the limits of integration for the angle $\theta$ in cylindrical coordinates?
In the application of Gauss's law to find the electric field around an infinite wire with uniform charge density, which statement is true regarding the limits of integration for the angle $\theta$ in cylindrical coordinates?
In the context of Gauss's law, what is the purpose of considering the closed surface when calculating the electric flux?
In the context of Gauss's law, what is the purpose of considering the closed surface when calculating the electric flux?
What key conclusion can be drawn from Faraday's ice pail experiment regarding the nature of electric charge and conductors?
What key conclusion can be drawn from Faraday's ice pail experiment regarding the nature of electric charge and conductors?
How does electrostatic shielding, such as that provided by a Faraday cage, protect the interior from external electric fields?
How does electrostatic shielding, such as that provided by a Faraday cage, protect the interior from external electric fields?
How does the presence of an empty cavity within a conductor affect the electric field inside the cavity?
How does the presence of an empty cavity within a conductor affect the electric field inside the cavity?
Imagine applying Gauss's Law to a uniformly charged sphere. What is the utility of choosing a spherical Gaussian surface concentric with the charged sphere?
Imagine applying Gauss's Law to a uniformly charged sphere. What is the utility of choosing a spherical Gaussian surface concentric with the charged sphere?
What is the electric field just beneath the surface of the conductor, where there is a negative surface charge density?
What is the electric field just beneath the surface of the conductor, where there is a negative surface charge density?
For which of the following charge distributions would Gauss's Law not be useful?
For which of the following charge distributions would Gauss's Law not be useful?
Consider a uniformly charged insulating sphere. How does the electric field within the sphere change with distance ( r ) from the center?
Consider a uniformly charged insulating sphere. How does the electric field within the sphere change with distance ( r ) from the center?
If Gauss's Law is expressed as $ \Phi = \oint_S \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$, what describes $ \epsilon_0$?
If Gauss's Law is expressed as $ \Phi = \oint_S \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$, what describes $ \epsilon_0$?
Why would knowledge of electric flux be useful?
Why would knowledge of electric flux be useful?
How does flux depend on the size of the sphere?
How does flux depend on the size of the sphere?
An imaginary cube of side L is in an electrical field (E). What is needed to find the flux?
An imaginary cube of side L is in an electrical field (E). What is needed to find the flux?
Does doubling the size of a 'box' (a closed surface) change flux?
Does doubling the size of a 'box' (a closed surface) change flux?
Flashcards
Gauss' Law application
Gauss' Law application
Electric field at a surface can be used to find charge within.
Electric Flux
Electric Flux
Measure of electric field through a surface.
Positive Charge Flux
Positive Charge Flux
Positive charge produces outward electric flux.
Negative Charge Flux
Negative Charge Flux
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Zero Net Charge
Zero Net Charge
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Charge and Flux
Charge and Flux
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Changing Box Size
Changing Box Size
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Electric Flux Formula
Electric Flux Formula
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Surface Edge-On
Surface Edge-On
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Gauss' Law Equation
Gauss' Law Equation
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Gaussian Surface
Gaussian Surface
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E · dA
E · dA
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Closed Surface Integral
Closed Surface Integral
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Gauss' Law: use case 1
Gauss' Law: use case 1
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Gauss' Law: use case 2
Gauss' Law: use case 2
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Gauss' Law: use case 3
Gauss' Law: use case 3
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Flux Through Sphere
Flux Through Sphere
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Flux and Enclosed Charge
Flux and Enclosed Charge
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Positive Flux
Positive Flux
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Negative Flux
Negative Flux
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Equal Flux
Equal Flux
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Charge on Conductor
Charge on Conductor
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E Field in Conductor
E Field in Conductor
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Field of a Line Charge
Field of a Line Charge
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Field of a Sheet
Field of a Sheet
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Faraday Cage
Faraday Cage
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Study Notes
Gauss' Law
- This lecture aims to demonstrate how to use electric fields to determine charge, to explain and calculate electric flux, to relate electric flux and charge, to calculate electric fields using Gauss's law, and to locate charge on a conductor.
Positive Charge and Electric Flux
- Positive charge inside a box results in outward electric flux.
- More charge corresponds to more flux.
Negative Charge and Electric Flux
- Negative charge produces inward flux.
- More negative charge leads to more inward flux.
Zero Net Charge
- With zero net charge inside a box, there are three scenarios to consider.
- There is no electric flux through the surface.
Factors Affecting Flux
- Doubling the charge within a box doubles the flux.
- Doubling the size of the box doesn't change the flux.
Uniform Electric Fields and Electric Flux Units
- For a uniform electric field in space, the electric flux is denoted as Ф= È-Ã = EA cos(0°).
- The units for electric flux [Ф] are N/C • m² or Nm²/C.
Calculating Electric Flux in Uniform Fields
- When a surface is face-on to an electric field, E and A are parallel, the angle between them is 0.
- The electric flux is ФE = È·Ã = EA.
- When the surface is tilted at an angle ф, the electric flux is ФE = È·Ã = EA cos ф.
- When the surface is edge-on to an electric field, E and A are perpendicular, and the angle is 90°.
- The electric flux is ФE = È·Ã = EA cos 90° = 0.
Electric Flux Example
- A disk of radius 0.10 m has a vector n at 30 degrees to E, and an electric field magnitude of 2.0 x 103 N/C.
- Calculation of the electric flux: Ф = E·A = EA cos(30°), where A = πr² = 0.0314 m².
- The electric flux is Ф = 54 Nm²/C.
Electric Flux Through a Cube
- An imaginary cube of side L in a uniform electric field E needs its flux calculated for each of its sides.
Electric Flux Through a Sphere
- The electric flux through a sphere is defined by the integral Ф = ∫ E·dA.
- Where E = kq/r² = 1/(4πε₀) q/r² and is parallel to dA everywhere on the surface.
- The electric flux simplifies to Ф = ∫ E·dA = E ∫ dA = EA.
- For a charge q = +3.0nC, the flux through a sphere of radius r=.20 m can be calculated.
Gauss' Law
- Gauss' Law: Ф = ∫ E·dA = qenc / ε₀.
- Electric flux produced by a charge can be calculated by integrating over a closed 2D surface.
- E field is a vector
- Dot product calculation: E·dA = ExdAx + EydAy + EzdAz = |E||dA|cosθ, with dA being an infinitesimal area element vector.
- The formula uses the total amount of charge but only counts the charge inside the closed surface S.
- Epsilon is the electrical permittivity of free space
Uses for Gauss’ Law
- To find the electric flux through a surface enclosing a charge.
- To find the total charge enclosed by a surface.
- To find the E field for highly symmetric distributions.
Gauss’ Law for Spherical Surfaces
- Flux through the sphere remains constant regardless of the sphere's size.
- Flux is solely dependent on the enclosed charge.
Gauss' Law with Non-Spherical Surfaces
- Flux remains independent of the surface and depends only on enclosed charge.
Positive and Negative Electric Flux
- Positive electric flux is associated with enclosed positive charge.
- Negative electric flux occurs if the charge is negative.
Spherical Gaussian Surface
- A spherical Gaussian surface #1 encloses and is centered on a point charge +q.
- A second spherical Gaussian surface #2 of the same size also encloses the charge but is not centered on it.
- The electric flux through both surfaces is the same.
Conceptual Example: Flux
- Flux A = +q/ε₀
- Flux B = -q/ε₀
- Flux C = 0
- Flux D = 0
- Net electric flux through surface C and surface D equal zero
Applications of Gauss' Law
- Under electrostatic conditions, excess charge resides on a conductor's surface.
- The electric field inside a conductor under electrostatic conditions is zero.
- The field outside a spherical conductor looks like a point charge under electrostatic conditions.
Field of a Line Charge
- Calculating E around an infinite positive wire with charge density λ involves using a closed cylindrical Gaussian surface.
- Electric field E is orthogonal to dA at the end caps.
- E is parallel(radially outwards) to dA on the cylinder.
- E is constant in value everywhere on the cylinder at a distance r from the wire.
- Integrating over the area involves two dimensions: dA = (rdθ) dl.
- The limits of integration are: dθ goes from 0 to 2π; dl goes from 0 to l (length of cylinder).
- Flux = E x Surface area
- Q(enc) = (charge density) x (length) = λ 1
- Gauss' Law gives us the flux = E(2πr) l = q/ε₀ = (λ l) /ε₀
Field of a Sheet of Charge
- Calculation of the constant value of E involves the integration of E dot dA.
Charges on Conductors with Cavities
- E = 0 inside a conductor
- An empty cavity inside a conductor has no electric field and no charge on its inner surface.
- Isolated charges inside a cavity induce opposite charge, canceling the field inside the conductor.
- This phenomenon was observed during Faraday's icepail experiment, which confirmed Gauss's Law.
Van de Graaff Generator
- The Van de Graaff generator operates on the same principle as Faraday's icepail experiment.
Electrostatic Shielding
- A conducting box (a Faraday cage) shields from electric fields.
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