Show that the retarded scalar potential given by

and the retarded vector potential given by

,

satisfy the Lorentz gauge condition.

A plane electromagnetic wave propagating in z-direction is incident on the boundary between two linear dielectric media. The boundary is in the x-y plane perpendicular to the direction of propagation of the wave. Using the boundary conditions on the electric and magnetic fields associated with the wave, show that if the incident wave is plane polarised then the reflected and transmitted waves also have the same polarisation.

Using Fourier integral theorem, obtain the expression for the velocity of a wave packet in a medium for which the dispersion relation is given as 

Using Maxwell's equations, obtain the boundary conditions for D, E, B, and at the interface of two dielectric media.

 At time t= 0, a particle having charge q is placed at rest at the point (0, 0) in a region of crossed uniform static electric and magnetic fields specified by Describe the motion of the particle and show that the motion is periodic.

Obtain the expressions for inhomogeneous Maxwell's equations in terms of scalar and vector potentials.

Obtain expressions for electric and magnetic fields in terms of scalar and vector potentials using homogeneous Maxwell's equations.

Explain the concept of displacement current. An infinitely long perfectly conducting cylindrical wire of radius a is surrounded by a coaxial perfectly conducting cylindrical tube of radius b. The potential difference between the wire and the tube is V(t). Compute V x B in the region between the wire and the tube.

A circular wire loop of radius ris kept in the x-y plane in a region of uniform magnetic field pointing in the z-direction. The magnitude of the magnetic field is held constant at B=0 from time t=0 to time.t=t₁ Thereafter, the magnetic field is increased at constant rate from B=0at time t=t₂ to BB till time f-12 and then reduced at a constant rate from B= B_o to B=0 at time t=t₃. a) Compute the magnetic flux through the circular wire loop as a function of time. b) Compute the induced emf in the circular loop as a function of time. c) Plot the magnetic flux and induced emf as a function of time.

k के वे मान ज्ञात कीजिए जिनके लिए सदिश

रैखिक स्वतंत्र है।

निम्नलिखित LPP को ग्राफीय विधि से हल कीजिए:

z= 10x₁ + 10x₂   का अधिकतमीकरण कीजिए।

जबकि :     4x₁ + 3x₂ ≤ 12

                6x₁ +18x₂ ≤ 36

                 X₁,X₂ ≥ 0.

निम्नलिखित खेल का ग्राफीय विधि से हल कीजिए:

निम्नलिखित नियतन समस्या की LPP सूचित कीजिए :

लाभ अधिकतमीकरण की निम्नलिखित नियतन समस्या को हल कीजिए:

निम्नलिखित परिवहन समस्या की (LPP) सूचित कीजिए :

निम्नलिखित रैखिक प्रोग्रामिंग (एलपी) समस्या को एकल विधि का उपयोग करके हल करें:

z = 6x₁ + 4x₂ का अधिकतमीकरण कीजिए

जबकि       2x₁ + 3x₂ ≤ 30

                 3x₁ + 2x₂ ≤ 24

                  x₁ + x₂ ≥ 3

                  x₁ , x₂ ≥ 0

k के किस मान के लिए निम्नलिखित सदिश, रैखिकतः स्वतंत्र हैं?

जाँच कीजिए कि निम्नलिखित समुच्चय अवमुख है :

न्यूनतम आव्यूह विधि का प्रयोग करके निम्नलिखित परिवहन समस्या का प्रारम्भिक आधारी सुसंगत हल ज्ञात कीजिए :

इस प्रकार इष्टतम हल ज्ञात कीजिए।

निम्नलिखित भुगतान आव्यूह में दिए गए शून्य-योग खेल को एक समतुल्य रैखिक प्रोग्रामन समस्या में परिवर्तित कीजिए: