TRB PG Assistant - Coulomb's law, Gauss's law, Poisson's and Laplace equations - boundary value problems, electrostatics of dielectric media, molecular polarizability and applications Online Test (MCQ Quiz)

1. What is the unit of electric charge according to Coulomb's law?

A) Volt B) Ampere C) Coulomb D) Ohm

2. 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?

A) Newton's Law of Universal Gravitation B) Hooke's Law C) Ohm's Law D) Boyle's Law

3. The constant of proportionality in Coulomb's law in vacuum is denoted by 'k'. What is its approximate value?

A) 8.988 x 10^9 N m^2/C^2 B) 6.674 x 10^-11 N m^2/kg^2 C) 1.602 x 10^-19 C D) 3.141 x 10^0

4. What is the mathematical expression for Coulomb's law for two point charges q1 and q2 separated by a distance r in vacuum?

A) F = k * |q1 * q2| / r B) F = k * |q1 * q2| / r^2 C) F = k * (q1 + q2) / r^2 D) F = k * sqrt(|q1 * q2|) / r

5. 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?

A) ∮ E ⋅ dA = Q_enc / ε₀ B) ∮ B ⋅ dA = 0 C) ∮ E ⋅ dl = -dΦ_B/dt D) ∮ B ⋅ dl = μ₀ * I_enc

6. In Gauss's law, ε₀ represents:

A) The permittivity of the medium B) The permeability of the medium C) The permittivity of free space D) The charge density

7. Gauss's law is particularly useful for calculating the electric field in situations with high degree of:

A) Asymmetry B) Symmetry C) Randomness D) Non-uniformity

8. Which of the following charge distributions would make Gauss's law most convenient for finding the electric field?

A) A single point charge B) Two point charges C) An infinitely long uniformly charged rod D) A randomly distributed set of charges

9. What is the divergence of the electric field in free space, according to Gauss's law in differential form?

A) ∇ ⋅ E = 0 B) ∇ ⋅ E = ρ / ε₀ C) ∇ × E = 0 D) ∇ × E = -∂B/∂t

10. Poisson's equation relates the electric potential (V) to the charge density (ρ) in a region.

A) ∇²V = 0 B) ∇²V = -ρ / ε₀ C) ∇²V = ρ / ε₀ D) ∇²V = -ε₀ / ρ

11. Laplace's equation is a special case of Poisson's equation. When does Poisson's equation reduce to Laplace's equation?

A) When the charge density is non-zero B) When the potential is zero C) When there are no free charges in the region (ρ = 0) D) When the electric field is uniform

12. 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?

A) Ampere's Law and Faraday's Law B) Coulomb's Law and Biot-Savart Law C) Gauss's Law, Poisson's Equation, and Laplace's Equation D) Ohm's Law and Kirchhoff's Laws

13. What is the physical meaning of the term 'boundary value problem' in electrostatics?

A) Solving equations only where charges exist B) Finding the solution of electrostatic equations that satisfies conditions on the boundaries of the region of interest C) Determining the value of the charge at the boundary D) Calculating the electric field along a boundary line

14. Electrostatics of dielectric media involves the study of how materials respond to electric fields. What is a dielectric material?

A) A material that conducts electricity very well B) A material that does not allow electric current to flow C) A material that can be easily polarized by an electric field but does not conduct charge D) A material that generates its own electric field

15. When a dielectric material is placed in an external electric field, what happens at the molecular level?

A) The molecules align randomly B) The molecules become ionized and conduct charge C) The molecules or their constituent charges are slightly displaced, leading to polarization D) The material heats up significantly

16. What is the dielectric constant (or relative permittivity) of a material?

A) The ratio of the electric field in vacuum to the electric field in the material B) The ratio of the electric field in the material to the electric field in vacuum C) The ratio of the capacitance of a capacitor with the dielectric to its capacitance without the dielectric D) The product of the electric field and the material's permittivity

17. The presence of a dielectric material between the plates of a parallel-plate capacitor generally:

A) Decreases its capacitance B) Increases its capacitance C) Does not affect its capacitance D) Makes it a conductor

18. Molecular polarizability (α) quantifies:

A) The ability of a material to conduct electricity B) The ease with which the electron cloud of an atom or molecule can be distorted by an external electric field C) The total charge density of a molecule D) The natural frequency of oscillation of a molecule

19. For a linear, isotropic dielectric, the induced dipole moment (p) is proportional to the applied electric field (E). The constant of proportionality is:

A) Permittivity of free space (ε₀) B) Molecular polarizability (α) C) Dielectric constant (ε_r) D) Charge density (ρ)

20. The relationship between the induced dipole moment (p), molecular polarizability (α), and the electric field (E) is given by:

A) p = α / E B) p = E / α C) p = α * E D) p = α + E

21. In a dielectric medium, the total electric field is reduced compared to the field in vacuum due to:

A) Increased charge density B) Induced polarization charge creating an opposing field C) The material becoming a conductor D) A decrease in molecular polarizability

22. An application of electrostatics involving dielectric media is found in:

A) Electric motors B) Capacitors C) Inductors D) Resistors

23. What is the primary function of a dielectric material in a capacitor?

A) To increase the distance between the plates B) To reduce the voltage rating of the capacitor C) To increase the capacitance by allowing closer plate spacing or reducing the effective electric field D) To make the capacitor a better conductor

24. Consider a spherical charge distribution with total charge Q. Using Gauss's law, what is the electric field outside the sphere (r > R)?

A) E = (1/4πε₀) * (Q/r) B) E = (1/4πε₀) * (Q/r^2) C) E = (1/4πε₀) * (Qr^2) D) E = 0

25. For a point charge q located at the origin, the electric potential V at a distance r is given by:

A) V = (1/4πε₀) * (q/r) B) V = (1/4πε₀) * (qr) C) V = (1/4πε₀) * (q/r^2) D) V = (1/4πε₀) * (q^2/r)

26. If the electric field is conservative, then the line integral of the electric field around any closed loop is:

A) Non-zero B) Equal to the charge enclosed C) Zero D) Equal to the flux

27. The relationship between electric field (E) and electric potential (V) is given by:

A) E = -∇V B) E = ∇V C) E = -∫V dl D) E = ∇⋅V

28. Poisson's equation ∇²V = -ρ/ε₀ is a second-order partial differential equation. What does ∇² represent?

A) The gradient operator B) The divergence operator C) The curl operator D) The Laplacian operator (divergence of the gradient)

29. In a region with no free charges, Laplace's equation ∇²V = 0 implies that:

A) The electric field is constant B) The electric potential has no maximum or minimum values within the region C) The electric potential is always zero D) The electric field is always zero

30. 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:

A) Uniform throughout the surface B) Zero C) Equal to the potential of the surrounding medium D) Proportional to the surface charge density

31. When a dielectric material is inserted into a capacitor, the electric field between the plates (for a constant charge on the plates) is:

A) Increased B) Decreased C) Unchanged D) Zero

32. What is 'dielectric strength'?

A) The maximum electric field a dielectric can withstand before breaking down and becoming conductive B) The measure of how easily a dielectric can be polarized C) The dielectric constant of the material D) The energy stored per unit volume in the dielectric

33. In the context of molecular polarizability, what is a 'dipole moment'?

A) A measure of the electric field strength B) A measure of the separation of positive and negative charges in a system C) The total charge of a molecule D) The net force on a molecule

34. Which type of dielectric material has its molecules possessing a permanent dipole moment even in the absence of an external electric field?

A) Non-polar dielectric B) Polar dielectric C) Conductor D) Insulator

35. When a non-polar dielectric is subjected to an electric field, the induced dipole moment arises from:

A) Rotation of permanent dipoles B) Alignment of existing dipoles C) Displacement of electron clouds relative to nuclei D) Generation of free electrons

36. 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?

A) D = ε₀E B) D = ε_rE C) D = ε₀ε_rE D) D = E / (ε₀ε_r)

37. 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?

A) ∮ D ⋅ dA = Q_free_enc B) ∮ E ⋅ dA = Q_total_enc / ε₀ C) ∮ D ⋅ dA = Q_total_enc D) ∮ ∇ ⋅ D dA = Q_free_enc

38. What is the unit of electric flux?

A) Newton per Coulomb B) Volt-meter C) Coulomb per square meter D) Newton meter squared per Coulomb

39. If a spherical Gaussian surface encloses a net charge of zero, the net electric flux through the surface is:

A) Zero B) Dependent on the radius of the sphere C) Infinite D) Equal to the permittivity of free space

40. 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:

A) Maximum-minimum principle B) Uniqueness theorem C) Superposition principle D) Gauss's theorem

41. 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?

A) It increases by a factor of ε_r B) It decreases by a factor of ε_r C) It remains unchanged D) It becomes zero

42. What is the unit of electric potential?

A) Joule B) Watt C) Volt D) Ampere

43. The process of inducing a dipole moment in a non-polar molecule by an external electric field is called:

A) Ionization B) Electrolysis C) Electronic polarization D) Ferroelectricity

44. 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?

A) Coulomb's Law B) Gauss's Law C) Poisson's Equation D) Laplace's Equation

45. The electric susceptibility (χ_e) of a dielectric material is related to its molecular polarizability (α) and the number density (n) of molecules by:

A) χ_e = nα / ε₀ B) χ_e = nαε₀ C) χ_e = n / (αε₀) D) χ_e = nε₀ / α

46. What is the relationship between the relative permittivity (ε_r) and electric susceptibility (χ_e)?

A) ε_r = 1 - χ_e B) ε_r = 1 + χ_e C) ε_r = χ_e - 1 D) ε_r = 1 / χ_e

47. The electric field inside a conductor in electrostatic equilibrium is:

A) Maximum at the center B) Non-zero and uniform C) Zero D) Equal to the external field

48. Gauss's law is particularly useful for calculating electric fields of charge distributions with:

A) No symmetry B) Spherical, cylindrical, or planar symmetry C) Complex, irregular shapes D) Varying charge density

49. If we consider a boundary between two dielectric media with different permittivities, the boundary conditions typically involve the continuity of:

A) The normal component of E and the tangential component of D B) The tangential component of E and the normal component of D C) Both tangential components E and D D) Both normal components E and D

50. The concept of 'molecular polarizability' is crucial for understanding the behavior of:

A) Conductors in electric fields B) Dielectric materials in electric fields C) Magnetic materials in magnetic fields D) Superconductors

Coulomb's law, Gauss's law, Poisson's and Laplace equations - boundary value problems, electrostatics of dielectric media, molecular polarizability and applications - Online Test

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