Calculus – JEE First

Biot-Savart law on a polygon

September 30th, 2022jeefirst0 Comments

A current $I$ flows along a thin wire, shaped as a regular polygon with $N$ sides, which can be inscribed intro a circle of radius $R$ . Find the magnetic field at the center of the polygon. What happens if $N$ is made very large?

Solution:

The polygon described in the problem is illustrated in the figure below, for $N=8$ . The magnetic field created by this structure is the superposition of fields from $N$ straight current carrying wires of length $2 R \sin (\pi/N)$ . Therefore, we will need to find the magnetic field due to a wire of finite length first.

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The Biot-Savart law gives the magnetic field …

Progressive refraction through glass layers

March 4th, 2022jeefirst0 Comments

JEE Advanced 2017 Paper 1, Question 10

A monochromatic light is traveling in a medium of refractive index $n=1.6$ . It enters a stack of glass layers from the bottom side at an angle $\theta=30^{\circ}$ . The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as $n_{m}=n-m \Delta n$ , where $n_{m}$ is the refractive index of the $m^{\rm th}$ slab and $\Delta n=0.1$ (see the figure). The ray is refracted out parallel to the interface between the $(m-1)^{\rm th}$ and $m^{\rm th}$ slabs from the right side of the stack. What is the value of $m$ ?

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Solution

Consider the trajectory of …

Terminal velocity in a magnetic field

February 23rd, 2022jeefirst0 Comments

A copper connector of mass $m$ slides down two smooth copper bars, set at an angle $\alpha$ to the horizontal, due to gravity (see figure). At the top the bars are interconnected through a resistance $R$ . The separation between the bars is $\ell$ . The system is located in a uniform magnetic field of induction $B$ , perpendicular to the plane in which the connector slides. The resistance of the bars, the connector and the sliding contacts, as well as the self-inductance of the loop are assumed to be negilible. If the connector is released from rest at $t=0$ ,

Find the velocty $v(t)$ of

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Missing energy in a rope and a capacitor

February 14th, 2022jeefirst0 Comments

Consider a uniform rope of mass density $\lambda$ coiled on a smooth horizontal table. One end is pulled straight up with a constant speed $v_0$ as shown.

Find the force exerted on the end of the rope as function of the height $y$ .
Compare the power delivered to the rope with the rate of change of the rope’s mechanical energy.

(This is a problem from chapter 5 of Kleppner and Kolenkow)

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To find the force exerted at the top end, note that if we were to pull up a fixed mass with constant velocity $v_0$ , the total force on the mass should …

Sliding on a block with a circular cut

January 30th, 2022jeefirst0 Comments

JEE Advanced 2017 Paper 1, Question 2

A block of mass $M$ has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at $x=0$ , in a co-ordinate system fixed to the table. A point mass $m$ is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block, its position is $x$ and the velocity is $v$ . At that instant, which of the following options is/are correct?

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The position

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