Gauss's Law and Electric Fields Quiz

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Questions and Answers

What condition must be satisfied for an electric field to be considered an electrostatic field?

The curl of the electric field must be positive.

The divergence of the electric field must be zero.

The curl of the electric field must be zero. (correct)

The electric field must be uniform.

What does the line integral of the electric field E⃗ from point a to point b represent?

It results in different values depending on the electric field strength.

It is path-dependent and varies with the trajectory taken.

It is independent of the path taken between the two points. (correct)

It is equal to the work done by the electric field.

Which of the following statements about the electric potential V(⃗r) is true?

It is defined as the integral of E⃗ over a closed loop.

It can be defined only when the electric field is not uniform.

It is defined as the negative line integral of E⃗ from a reference point O to point r. (correct)

It is dependent on the path taken from point O to r.

If the curl of an electric field vector E⃗ is non-zero, which of the following can be inferred?

The electric field is not electrostatic. Signup and view all the answers

What does the expression ∇⃗ × E⃗ = 0 signify about the electric field E⃗?

The electric field can be represented as a conservative vector field. Signup and view all the answers

Which statement accurately describes a valid electrostatic field?

The curl of the electric field is zero. Signup and view all the answers

What does the principle of superposition state regarding electric fields?

The total field is a vector sum of the individual fields. Signup and view all the answers

In the application of Stoke's theorem to the electric field, which outcome is indicated when the curl of the electric field is zero?

The work done in moving a charge around a closed loop is zero. Signup and view all the answers

Which scenario would suggest that an electrostatic field is invalid?

A field where the curl is dependent on the position. Signup and view all the answers

If given an electric field expression, how would you determine if it is a valid electrostatic field?

Evaluate if the curl of the field is equal to zero. Signup and view all the answers

What is the direction of the electric field E⃗ above an infinite plane of charge?

Upwards away from the plane Signup and view all the answers

According to Gauss's law, how is the electric field E related to charge density σ and area A?

E = σ/(2ϵ0) Signup and view all the answers

What explains the independence of the electric field from the distance for an infinite plane?

More charge comes into the field of view as distance increases. Signup and view all the answers

If the charge of an infinite plane is uniform, how does its electric field change if the area A is doubled?

The electric field E remains the same. Signup and view all the answers

In Example 6, what is the correct formula for the electric field E⃗ due to the uniformly charged solid sphere inside of radius R?

E⃗ = σ/(2ϵ0) Signup and view all the answers

What factor does the 'n' denote in the context of the electric field E⃗ for a surface chosen in a specific direction?

The orientation of the surface normal. Signup and view all the answers

Which law explains the relationship between electric fields and charge distributions?

Gauss's Law Signup and view all the answers

When calculating the electric field E of an infinite plane, which aspect does NOT affect the electric field value?

Distance from the plane Signup and view all the answers

What determines the form of the electric field outside a uniformly charged solid sphere?

The total charge and distance from the center Signup and view all the answers

According to Gauss's law, how is the electric field related to the total charge enclosed by a Gaussian surface?

The electric field is proportional to the charge enclosed and inversely proportional to the area of the Gaussian surface. Signup and view all the answers

What is the expression for the magnitude of the electric field outside a uniformly charged solid sphere?

$E = \frac{q}{4\pi\epsilon_0 r}$ Signup and view all the answers

Why is the electric field expressed in the form $E = E_{rb}$ outside a uniformly charged solid sphere?

Because the electric field has radial symmetry. Signup and view all the answers

In the context of a charged cylinder, what does the charge density being proportional to the distance from the axis imply?

The charge density increases linearly as you move away from the center. Signup and view all the answers

Which of the following is a correct application of Gauss's law in finding the electric field?

Choosing a Gaussian surface that aligns with the symmetry of the charge distribution. Signup and view all the answers

What condition must be met for the application of Gauss's law to be straightforward?

The electric field must have uniform direction throughout the Gaussian surface. Signup and view all the answers

When applying Gauss's law to a uniformly charged solid sphere, what represents $Q_{enc}$?

The total charge within the sphere's volume. Signup and view all the answers

What is the expected outcome on the force experienced by a test charge Q at the center of a regular 13-sided polygon with 13 equal charges q placed at its corners?

The force is zero due to symmetry. Signup and view all the answers

What happens to the force on the test charge Q if one of the charges q at a corner of the polygon is removed?

The force increases towards the opposite side. Signup and view all the answers

How would you derive the electric field a distance z above the midpoint between two equal but opposite charges?

By summing the field contributions from each charge. Signup and view all the answers

What would the electric field at a distance z above one end of a uniform line charge of length L be expected to approach as z becomes much larger than L?

It approaches the field of a point charge. Signup and view all the answers

For a flat circular disk of radius R carrying uniform surface charge σ, what happens in the limit when R approaches infinity?

The electric field becomes uniform and constant. Signup and view all the answers

In analyzing the charge density ρ given the electric field E⃗ = kr^3 rb in spherical coordinates, what is the first step to find ρ?

Use the divergence of E to find charge density. Signup and view all the answers

What would the total charge contained in a sphere of radius R centered at the origin be for a charge density derived from E⃗ = kr^3 rb?

It increases exponentially with R. Signup and view all the answers

What is the electric field configuration when analyzing two infinite parallel planes with equal but opposite charge densities ±σ?

The electric field points toward the positively charged plane in all regions. Signup and view all the answers

Study Notes

Gauss's Law in Differential Form

Gauss's Law in differential form relates the divergence of the electric field to the charge density.
The equation is: ∇ · E⃗ = ρ / ϵ0
Where:
- ∇ is the divergence operator
- E⃗ is the electric field
- ρ is the charge density
- ϵ0 is the permittivity of free space

Example 4: Electric Field Outside a Charged Sphere

The problem involves finding the electric field outside a uniformly charged solid sphere with radius R and total charge q.
Gauss's Law states: ∫S E⃗ · d⃗a = Qenc / ϵ0
For a spherical Gaussian surface with radius r > R, the enclosed charge (Qenc) is equal to q.
Due to spherical symmetry, the electric field is radial and has a constant magnitude.
Therefore, E⃗ · d⃗a = E r^2 sinθ dθ dϕ, where E is the magnitude of the electric field.
Applying Gauss's Law and integrating over the spherical Gaussian surface, the electric field outside the charged sphere is calculated to be E = q / (4πϵ0r^2).

Example 5: Electric Field of a Charged Cylinder

The problem involves a long cylinder with a charge density proportional to the distance from the axis, ρ = ks, where k is a constant.
Due to cylindrical symmetry, the electric field points radially away from the axis.
Through integration, the electric field can be determined to be E = ks^2 / (3ϵ0).

Example 6: Electric Field of an Infinite Plane

The problem involves finding the electric field a distance z above an infinite plane with a uniform surface charge density σ.
Using a Gaussian surface (a cylinder that intersects the plane) and Gauss's Law, the electric field is found to be independent of the distance z.
E = σ / (2ϵ0), where E is the magnitude of the electric field.

Curl of the Electric Field

The curl of an electrostatic field is always zero.
This implies that the electric field can be expressed as the gradient of a scalar potential, E⃗ = -∇V.
This relationship holds because the line integral of the electric field around any closed loop is always zero.

Electric Potential

The electric potential, denoted by V, is a scalar function that describes the electric potential energy per unit charge at a given point.
It is defined as the negative line integral of the electric field from a reference point to the point in question, V(⃗r) = -∫O→⃗r E⃗ · d ⃗l.
The electric potential is independent of the path taken, only depending on the initial and final points.

Problems for Homework

Problem 4: Calculate the electric field at a distance z above the midpoint between two equal but opposite charges, q and -q, separated by distance d.
Problem 5: Calculate the electric field at a distance z above one end of a straight line segment of length L with a uniform line charge λ.
Problem 6: Calculate the electric field at a distance z above the center of a square loop with side a carrying a uniform line charge λ.
Problem 7: Calculate the electric field at a distance z above the center of a circular loop with radius r carrying a uniform line charge λ.
Problem 8: Calculate the electric field at a distance z above the center of a flat circular disk with radius R carrying a uniform surface charge σ.
Problem 9: Given an electric field E⃗ = kr^3 r̂ in spherical coordinates, find the charge density ρ and the total charge contained in a sphere of radius R centered at the origin.
Problem 10: Determine the flux of the electric field through a shaded side of a cube, with a charge q positioned at a back corner.
Problem 11: Two infinite parallel planes carry equal but opposite uniform charge densities ±σ. Find the field in each of the three regions: (i) to the left of both, (ii) between them, (iii) to the right of both.

Other Key Concepts

Superposition principle: The total electric field at a point due to multiple charges is the vector sum of the individual fields created by each charge.
Stoke's theorem: The line integral of a vector field around a closed loop is equal to the surface integral of the curl of the vector field over the enclosed surface.

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Description

Test your understanding of Gauss's Law in its differential form and its applications, particularly when calculating the electric field outside a charged sphere. Dive into the essential equations and concepts involved in electrostatics with specific focus on electric fields and charge density.