Coulomb's law, Gauss's law, Poisson's and Laplace equations - boundary value problems, electrostatics of dielectric media, molecular polarizability and applications - One Line Questions
1.
What is the divergence of the electric field in free space, according to Gauss's law in differential form? —
∇ ⋅ E = ρ / ε₀
2.
Poisson's equation relates the electric potential (V) to the charge density (ρ) in a region. —
∇²V = -ρ / ε₀
3.
Gauss's law in its most general form, applicable to dielectrics, is often written in terms of the electric displacement field D. What is this form? —
∮ D ⋅ dA = Q_free_enc
4.
Gauss's law relates the electric flux through a closed surface to the net electric charge enclosed within that surface. What is the mathematical formulation of Gauss's law? —
∮ E ⋅ dA = Q_enc / ε₀
5.
The constant of proportionality in Coulomb's law in vacuum is denoted by 'k'. What is its approximate value? —
8.988 x 10^9 N m^2/C^2
6.
Electrostatics of dielectric media involves the study of how materials respond to electric fields. What is a dielectric material? —
A material that can be easily polarized by an electric field but does not conduct charge
7.
In the context of molecular polarizability, what is a 'dipole moment'? —
A measure of the separation of positive and negative charges in a system
8.
Which of the following charge distributions would make Gauss's law most convenient for finding the electric field? —
An infinitely long uniformly charged rod
9.
The boundary value problem in electrostatics involves finding the electric potential or field that satisfies certain equations and boundary conditions. Which equations are fundamental to solving these problems? —
Gauss's Law, Poisson's Equation, and Laplace's Equation
10.
Gauss's law is particularly useful for calculating the electric field in situations with high degree of: —
Symmetry
11.
The concept of 'molecular polarizability' is crucial for understanding the behavior of: —
Dielectric materials in electric fields
12.
Which equation forms the basis for understanding the behavior of electric fields in the presence of conductors, where the electric field inside must be zero in electrostatics? —
Gauss's Law
13.
The electric displacement field D is defined as D = εE, where ε is the permittivity of the medium. In a linear dielectric, ε = ε₀ε_r. What is the relationship between D and E? —
D = ε₀ε_rE
14.
The presence of a dielectric material between the plates of a parallel-plate capacitor generally: —
Increases its capacitance
15.
The relationship between electric field (E) and electric potential (V) is given by: —
E = -∇V
16.
Consider a spherical charge distribution with total charge Q. Using Gauss's law, what is the electric field outside the sphere (r > R)? —
E = (1/4πε₀) * (Q/r^2)
17.
An application of electrostatics involving dielectric media is found in: —
Capacitors
18.
What is the mathematical expression for Coulomb's law for two point charges q1 and q2 separated by a distance r in vacuum? —
F = k * |q1 * q2| / r^2
19.
When a dielectric material is inserted into a capacitor, the electric field between the plates (for a constant charge on the plates) is: —
Decreased
20.
In a dielectric medium, the total electric field is reduced compared to the field in vacuum due to: —
Induced polarization charge creating an opposing field
21.
The process of inducing a dipole moment in a non-polar molecule by an external electric field is called: —
Electronic polarization
22.
Consider a parallel-plate capacitor with a vacuum between the plates. If a dielectric slab of thickness 'd' (equal to the plate separation) and dielectric constant ε_r is inserted, and the charge Q on the plates is kept constant, how does the electric field change? —
It decreases by a factor of ε_r
23.
What is the unit of electric potential? —
Volt
24.
The electric field inside a conductor in electrostatic equilibrium is: —
Zero
25.
The solution to Laplace's equation in a certain region is unique if the potential is specified on the entire boundary of that region. This is known as the: —
Uniqueness theorem
26.
What is the unit of electric flux? —
Volt-meter
27.
Coulomb's law states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This inverse square relationship is characteristic of which fundamental force law? —
Newton's Law of Universal Gravitation
28.
Gauss's law is particularly useful for calculating electric fields of charge distributions with: —
Spherical, cylindrical, or planar symmetry
29.
Which type of dielectric material has its molecules possessing a permanent dipole moment even in the absence of an external electric field? —
Polar dielectric
30.
If the electric field is conservative, then the line integral of the electric field around any closed loop is: —
Zero
31.
The relationship between the induced dipole moment (p), molecular polarizability (α), and the electric field (E) is given by: —
p = α * E
32.
For a linear, isotropic dielectric, the induced dipole moment (p) is proportional to the applied electric field (E). The constant of proportionality is: —
Molecular polarizability (α)
33.
When a non-polar dielectric is subjected to an electric field, the induced dipole moment arises from: —
Displacement of electron clouds relative to nuclei
34.
What is the physical meaning of the term 'boundary value problem' in electrostatics? —
Finding the solution of electrostatic equations that satisfies conditions on the boundaries of the region of interest
35.
Molecular polarizability (α) quantifies: —
The ease with which the electron cloud of an atom or molecule can be distorted by an external electric field
36.
In a region with no free charges, Laplace's equation ∇²V = 0 implies that: —
The electric potential has no maximum or minimum values within the region
37.
Poisson's equation ∇²V = -ρ/ε₀ is a second-order partial differential equation. What does ∇² represent? —
The Laplacian operator (divergence of the gradient)
38.
What is 'dielectric strength'? —
The maximum electric field a dielectric can withstand before breaking down and becoming conductive
39.
When a dielectric material is placed in an external electric field, what happens at the molecular level? —
The molecules or their constituent charges are slightly displaced, leading to polarization
40.
If we consider a boundary between two dielectric media with different permittivities, the boundary conditions typically involve the continuity of: —
The tangential component of E and the normal component of D
41.
In Gauss's law, ε₀ represents: —
The permittivity of free space
42.
What is the dielectric constant (or relative permittivity) of a material? —
The ratio of the capacitance of a capacitor with the dielectric to its capacitance without the dielectric
43.
What is the primary function of a dielectric material in a capacitor? —
To increase the capacitance by allowing closer plate spacing or reducing the effective electric field
44.
Boundary conditions for electrostatic problems typically involve specifying the value of the electric potential or its normal derivative on the boundaries of the conductor or the region of interest. For an isolated conductor in vacuum, the potential on its surface is: —
Uniform throughout the surface
45.
For a point charge q located at the origin, the electric potential V at a distance r is given by: —
V = (1/4πε₀) * (q/r)
46.
What is the unit of electric charge according to Coulomb's law? —
Coulomb
47.
Laplace's equation is a special case of Poisson's equation. When does Poisson's equation reduce to Laplace's equation? —
When there are no free charges in the region (ρ = 0)
48.
If a spherical Gaussian surface encloses a net charge of zero, the net electric flux through the surface is: —
Zero
49.
What is the relationship between the relative permittivity (ε_r) and electric susceptibility (χ_e)? —
ε_r = 1 + χ_e
50.
The electric susceptibility (χ_e) of a dielectric material is related to its molecular polarizability (α) and the number density (n) of molecules by: —
χ_e = nα / ε₀