Circular orbit in a harmonic potential

JEE Advanced 2018 Paper 1, Question 1

The potential energy of a particle of mass m at a distance r from a fixed point O is given by V(r)=k r^{2} / 2, where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius R about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?

  1. v=\sqrt{\frac{k}{2 m}} R
  2. v=\sqrt{\frac{k}{m}} R
  3. L=\sqrt{m k} R^{2}
  4. L=\sqrt{\frac{m k}{2}} R^{2}

Solution

The force due to the given potential is

(1)   \begin{equation*}   {\bf F} = -\frac{\partial V}{\partial r} \hat{\bf r} = -k r \hat{\bf r} . \end{equation*}

The mass also experiences a centrifugal force due to its circular motion,

(2)   \begin{equation*}   {\bf F}_{\rm cent} = \frac{m v^2}{r} \hat{\bf r} \end{equation*}

For the …

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Total internal reflection from a prism

JEE Advanced 2019 Paper 2, Question 11

A monochromatic light is incident from air on a refracting surface of a prism of angle 75^{\circ} and refractive index n_{0}=\sqrt{3}. The other refracting surface of the prism is coated by a thin film of material of refractive index n as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of \theta \leq 60^{\circ}. What is the value of n^{2}?

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Solution

First we trace the given incident ray through the prism to find the angle at which it is incident on the opposite face (see figure below). For …

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A spherical capacitor

A spherical conducting shell with radius b is concentric with a conducting ball with radius a, with a<b.

  1. Compute the capacitance C = Q / \Delta \phi when the shell is grounded and the ball has charge Q.
  2. Compute the capacitance when the ball is grounded and the shell has charge Q.
  3. Compute the full matrix of coefficients of capacitance for the two conductors.
  4. Considering these conductors as a capacitor, determine its capacitance. That is, assign equal and opposite charges \pm Q to the shell and the ball, and compute C = Q / \Delta \phi.

Related Problem: Insulating spherical shell with a hole

Solution

(a) First, we ground the shell and give the …

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Mirror on a spring

JEE Advanced 2019 Paper 2, Question 12

A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency \Omega such that \frac{4 \pi M \Omega}{h} = 10^{24} \, {\rm m}^{-2} with h as Planck’s constant. N photons of wavelength \lambda = 8 \pi \times 10^{-6} \, {\rm m} strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 \mu{\rm m}.
If the value of N is x \times 10^{12}, what is the value of x?

[Consider the spring as massless]

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Solution

The momentum carried by a single photon is given by the de Broglie relation p_{\rm ph} = h/\lambda. The total momentum carried by N photons is therefore N \times p_{\rm ph} = Nh/\lambda. All the photons hit the mirror simultaneously and are reflected …

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Dipole in a uniform electric field

JEE Advanced 2019 Paper 2, Question 4

An electric dipole with dipole moment \frac{p_{0}}{\sqrt{2}}(\hat{i}+\hat{j}) is held fixed at the origin O in the presence of an uniform electric field of magnitude E_{0}. If the potential is constant on a circle of radius R centered at the origin as shown in figure, then the correct statement(s) is/are:

(\varepsilon_{0} is permittivity of free space. R \gg dipole size)

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  1. R=\left(\frac{p_{0}}{4 \pi \epsilon_{0} E_{0}}\right)^{1 / 3}
  2. Total electric field at point A is {\bf E}^A=\sqrt{2} E_{0}(\hat{i}+\hat{j})
  3. Total electric field at point B is {\bf E}^B=0
  4. The magnitude of total electric field on any two points of the circle will be same.

Solution

The potential due to the dipole kept at the origin …

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