Conducting wire in a magnetic field

JEE Advanced 2019 Paper 1, Question 6

A conducting wire of parabolic shape, initially y = x^2, is moving with velocity \vec v = v_0 \hat i in a non-uniform magnetic field \vec B = B_0 \left( 1 + \left( \frac{y}{L} \right)^\beta \right) \hat k, as shown in the figure below. If v_0, B_0, L and \beta are positive constants and \Delta \phi is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:

Rendered by

  1. |\Delta \phi| = \frac{1}{2} B_0 v_0 L for \beta = 0.
  2. |\Delta \phi| = \frac{4}{3} B_0 v_0 L for \beta = 2.
  3. |\Delta \phi| remains the same if the parabolic wire is replaced by a straight wire, y=x initially, of length \sqrt{2} L.
  4. |\Delta \phi| is proportional to the length of the wire projected on the y axis.

Related Problems:
Electromagnetic induction in a twisted loop
Terminal velocity in a magnetic field

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Radioactive decay of Potassium

JEE Advanced 2019 Paper 1, Question 4

In a radioactive sample, {}^{40}_{19}{\rm K} nuclei decay into stable {}^{40}_{20}{\rm Ca} nuclei with decay constant 4.5 \times 10^{-10} per year or into stable {}^{40}_{18}{\rm Ar} nuclei with decay constant 0.5 \times 10^{-10} per year. In this sample all the stable {}^{40}_{20}{\rm Ca} and {}^{40}_{18}{\rm Ar} nuclei are produced by the {}^{40}_{19}{\rm K} nuclei only. In time t_1 \times 10^9 years, if the ratio of the sum of stable {}^{40}_{20}{\rm Ca} and {}^{40}_{18}{\rm Ar} nuclei to the radioactive {}^{40}_{19}{\rm K} nuclei is 99, the value of t_1 will be [Given \ln 10 = 2.3],

  1. 1.15
  2. 9.2
  3. 2.3
  4. 4.6


Let us first consider the situation where a sample of X nuclei decays into a single type of nuclei Y. The number N of X nuclei changes with …

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Rod heated by wire

JEE Advanced 2019 Paper 1, Question 3

A current carrying wire heats a metal rod. The wire provides a constant power P to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature T in the metal rod changes with time t as

(1)   \begin{equation*}  T(t) = T_0 (1 + \beta t^{\frac{1}{4}}) \end{equation*}

where \beta is a constant with appropriate dimension while T_0 is a constant with dimension of temperature. The heat capacity of the metal is,

  1. \frac{4 P [T(t) - T_0]^3}{\beta^4 T_0^4}
  2. \frac{4 P [T(t) - T_0]^4}{\beta^4 T_0^5}
  3. \frac{4 P [T(t) - T_0]^2}{\beta^4 T_0^3}
  4. \frac{4 P [T(t) - T_0]}{\beta^4 T_0^2}


The heat capacity of an object is defined by the relation

(2)   \begin{equation*}  \Delta Q = C \Delta T \end{equation*}

where \Delta Q is the heat that the object absorbs and \Delta T is the resulting temperature change of …

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Insulating spherical shell with a hole

JEE Advanced 2019 Paper 1, Question 2.

A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is V_0. A hole with a small area \alpha 4 \pi R^2 \ (\alpha \ll 1) is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

  1. The potential at the center of the shell is reduced by 2 \alpha V_0
  2. The magnitude of the eletric field at the center of the shell is reduced by \frac{\alpha V_0}{2 R}
  3. The ratio of the potential at the center of the shell to that of the point at R/2 from center towards
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Spherical gas cloud

JEE Advanced 2019 Paper 1, Question 1

Consider a spherical gas cloud of mass density \rho(r) in free space where r is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If \rho(r) is constant in time, the particle number density n(r) = \rho(r)/m is [G is the universal gravitational constant]

  1. \frac{K}{2 \pi r^2 m^2 G}
  2. \frac{K}{\pi r^2 m^2 G}
  3. \frac{3 K}{\pi r^2 m^2 G}
  4. \frac{K}{6 \pi r^2 m^2 G}


Consider a small parcel of gas in the cloud with mass \Delta m at a distance r from the center of the cloud (see figure). …

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