Trivia Questions On Differential Equation

When a capacitor is short-circuited, it essentially becomes a wire. In this case, the ohmmeter is measuring the resistance between the two terminals of the capacitor, which should ideally be zero for a wire. Therefore, the correct answer is zero.

The statement "The potential difference between two points is zero" does not state that the electrostatic field is conservative. This is because a conservative field implies that the work done in moving a charge between two points is independent of the path taken, not necessarily that the potential difference is zero. In other words, a conservative field means that the work done in a closed path is zero, but it does not necessarily mean that the potential difference between any two points is zero.

The induced electromotive force (e.m.f.) in a coil when a magnet is in motion relative to it does not depend on the resistance of the coil. The induced e.m.f. is determined by the rate at which the magnetic field lines passing through the coil change, which is influenced by factors such as the motion of the magnet and the number of turns of the coil. The resistance of the coil, on the other hand, affects the current that flows in the coil but does not directly impact the induced e.m.f.

The correct answer is "the field in which all lines of magnetic flux are parallel and equidistant." This means that in a uniform magnetic field, the magnetic field lines are evenly spaced and run parallel to each other. This type of field is often created by using a set of parallel conductors or by using a solenoid, which consists of a single conductor wound into a coil shape.

A singular solution refers to a solution of a differential equation that cannot be obtained from the general solution. It is a unique solution that does not fit into the general framework of solutions. Singular solutions often arise when there are special conditions or constraints in the equation that lead to unique solutions. These solutions may have different properties or behaviors compared to the general solutions, making them distinct and noteworthy.

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When a copper disc is rotated rapidly below a freely suspended magnetic needle, the magnetic needle experiences a phenomenon called electromagnetic induction. This phenomenon occurs because the rotating disc creates a changing magnetic field, which induces an electric current in the disc. According to Lenz's law, the induced current creates a magnetic field that opposes the change in the original magnetic field. As a result, the magnetic needle is repelled by the magnetic field created by the induced current, causing it to rotate in the same direction as the disc but with a lower velocity. Therefore, the correct answer is "less than that of disc and in the same direction."

A partial differential equation requires two or more independent variables because it involves the differentiation of a function with respect to multiple independent variables. In contrast, ordinary differential equations only involve differentiation with respect to a single independent variable. The presence of multiple independent variables in a partial differential equation allows for the consideration of more complex systems and phenomena that cannot be adequately described by ordinary differential equations.

According to coulomb's law, the force or field strength at point 'P' will act only in radial (R) direction while longitudinal components (L) will get cancelled out.

Vinyls is the correct answer because thermo-plastics are a type of plastic that can be melted and re-molded multiple times without undergoing any significant chemical change. Vinyls are a type of thermo-plastic that can be softened by heat and re-molded into various shapes. Epoxies, resins, and vulcanized rubber, on the other hand, are not thermo-plastics as they do not have the ability to be melted and re-molded multiple times.

The Laplace transform of cos(at) is (s/(s^2 + a^2)). This can be derived using the definition of the Laplace transform and the trigonometric identity cos(at) = (e^(iat) + e^(-iat))/2. By applying the linearity property and the transform of e^(at), the Laplace transform of cos(at) can be obtained as (s/(s^2 + a^2)).

In the Fourier series for odd functions, all the coefficients with the subscript "a" become zero. This is because odd functions have symmetry about the origin, which means that their Fourier series only contains sine terms and no cosine terms. The coefficients with the subscript "a" represent the cosine terms in the Fourier series, so in the case of odd functions, they are all zero. The coefficients with the subscript "b" represent the sine terms in the Fourier series and are not affected. The coefficient a0 represents the DC offset or the average value of the function, and it is also not affected by the odd symmetry.

Maxwell's Equations comprises of Gauss’s Faraday’s and Ampere’s equation which help studying the application of electromagnetic field to any medium. These equations can be represented in point as well as integral form.

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The correct answer is that the stored energy density for practicable field strength is low in the electric field. This means that the electric field is not able to store as much energy as the magnetic field can for a given field strength. Therefore, in electromechanical energy conversion, a magnetic field is employed as the medium instead of an electric field. This allows for more efficient energy conversion and storage.

A Laplace Transform exists when the function is both piece-wise continuous and of exponential order. Piece-wise continuity means that the function is continuous except for a finite number of discontinuities. Exponential order means that the function can be bounded by an exponential function for all values of t. Both conditions are necessary for the existence of a Laplace Transform.

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Aluminum alloy has an FCC (face-centered cubic) structure, which means that the aluminum atoms are arranged in a cubic lattice with additional atoms located at the center of each face of the cube. This arrangement allows for efficient packing of atoms and provides the alloy with good ductility, high strength, and excellent corrosion resistance. The FCC structure is commonly found in many metallic materials, including aluminum and its alloys, making the statement true.

Brittle materials typically fail fast and with little to no plastic deformation. Unlike ductile materials, which can undergo significant plastic deformation before failure, brittle materials fracture abruptly without significant deformation. Therefore, the statement is false.

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Ceramics are known for their high melting points, so the attribute "low melting point" does not apply to ceramics.

The expression L[f'(t)] represents the Laplace transform of the derivative of the function f(t). The Laplace transform of a derivative is given by s times the Laplace transform of the original function minus the initial value of the function. Therefore, the correct answer is S L[f(t)] - f(0), as it matches this formula.

The Wronskian of two linearly dependent functions is always zero. This is because linearly dependent functions can be expressed as a constant multiple of each other. When calculating the Wronskian using the formula W(f, g) = f g′ – g f ′, the resulting expression will be zero since the functions are proportional to each other. Therefore, the correct answer is zero.

Valence electrons are the outermost electrons in an atom and are responsible for the chemical behavior of an element. They determine how atoms bond with each other and form compounds, which in turn affects the material properties. Neutrons and protons are found in the nucleus of an atom and do not directly influence the chemical properties of an element. Electrons, in general, contribute to the overall properties of a material, but it is specifically the valence electrons that have the greatest impact on the material's characteristics.

In a simple cubic crystal structure, each atom is surrounded by 6 nearest neighboring atoms. Therefore, the coordination number, which represents the number of nearest neighbors for each atom, is 6.

The correct answer is "tan x - x + C" because the integral of tan x is equal to ln|sec x| + C, and the integral of x is equal to (1/2)x^2 + C. Therefore, the integral of tan x + x is equal to ln|sec x| + (1/2)x^2 + C. Since the given options only include tan x - x + C, it matches the correct answer.

Atomic packing factor refers to the volume fraction of atoms in a unit cell of a crystal structure. It is a measure of how efficiently the atoms are arranged in the crystal lattice. The higher the atomic packing factor, the more closely packed the atoms are in the unit cell. This factor is important in determining the density and physical properties of materials. The other options mentioned, such as the distance between two adjacent atoms and the projected area fraction of atoms on a plane, are not accurate definitions of atomic packing factor.