Solve your IGNOU Doubts

Question:

Two equal of mass m attached by a thread of length b are oriented vertically as shown in the figure. The acceleration decrease according to height as $g = g_0(1 - z / a)$ , where g₀ and a are constants. Calculate the vertical acceleration of the system as measured from the centre of the thread.

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Question:

Derive an integral equation corresponding to the ODE:

$$y'' - 2y = 0$
subject to the conditions: $y(0) = 4; y'(0) = -2$

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Question:

Reduce the following PDE into three ODEs:

$$\left[ \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right] f(x, y, z) + k^2 f(x, y, z) = 0$

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Question:

Two frictionless blocks of equal masses m are connected by a massless rigid rod of length L. The system is constrained to move in the vertical plane under the action of an applied force $\vec{F}$ as shown in the figure. Obtain the equation of motion of the system and calculate the condition for static equilibrium.

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Question:

What is meant by magnetic reluctance? Derive an expression for the reluctance of a composite magnetic circuit consisting of an iron core and an air gap. Clearly state the assumptions made.

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Question:

A long straight wire carries a steady current I. The wire is bent to form a circular arc of angle $\theta$ and radius R. Calculate the magnetic field at the centre of curvature and determine its direction using the right-hand rule.

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Question:

Define and explain the gyromagnetic ratio and Landé g-factor, stating their physical importance.

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Question:

Derive an expression for the magnetic moment associated with orbital motion of an electron in an atom.

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Question:

Obtain an expression for the magnetic vector potential, $\vec{A}$ due to a circular current loop of radius a carrying steady current I, at a point on its axis at distance x from the centre. Discuss the limiting case for $x \gg a$ .

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Question:

Define the electric displacement vector $\vec{D}$ . Starting from Maxwell's equations, deduce Gauss's law in dielectric media. Explain the physical significance of bound and free charges.

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Question:

calculate the force experienced by the charge q.

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Question:

Show that in the terms of the $|\uparrow\rangle$ and $|\downarrow\rangle$ basis vectors defined by
$\hat{S}_z|\uparrow\rangle = \frac{\hbar}{2}|\uparrow\rangle$ ; $\hat{S}_z|\downarrow\rangle = -\frac{\hbar}{2}|\downarrow\rangle$ , we can write: $\hat{S}_x = \frac{\hbar}{2} \left[ |\uparrow\rangle\langle\downarrow| + |\downarrow\rangle\langle\uparrow| \right]$

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Question:

i) Write down the angular momentum states $|j, m_j\rangle$ and calculate the matrix elements of $\hat{J}_x, \hat{J}_y$ and $\hat{J}_z$ for $j = 1/2$ .
ii) For the angular momentum state $|3,1\rangle$ show that $J_+|3,1\rangle = \hbar\sqrt{10}|3,2\rangle$ and $J_-|3,1\rangle = 2\hbar\sqrt{3}|3,0\rangle$ .

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Question:

determine the electrostatic potential in the region above the plane,

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Question:

An infinite plane conducting surface is held at zero potential. A point charge q is placed at a distance d above the plane. Using the method of images,

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Question:

Obtain Poisson's equation for electrostatics using the result derived in (a) above. Write down the general solution of Poisson's equation and explain its physical significance for a given charge distribution.

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Question:

For the simple harmonic oscillator
i) Show that $[\hat{H}, \hat{a}] = -\hbar\omega\hat{a}$ .
ii) Calculate the matrix element $\langle n|\hat{x}|n + 1\rangle$

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Question:

Starting from Coulomb's law, derive Gauss's law in differential form.

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Question:

Derive the Heisenberg equations of motion for $\hat{S}_x$ and $\hat{S}_y$ for the Hamiltonian $\hat{H} = \omega\hat{S}_z$ .

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Question:

A uniformly charged solid cylinder of radius a and length L carries a total charge Q. Assuming the observation point lies at a distance x from the centre of the cylinder along its axis and that $L \gg a$ , derive an expression for the electric field on the axis of the cylinder. Discuss the limiting case when $x \gg L$ .