Biot-Savart law on a polygon

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?


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 …

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Progressive refraction through glass layers

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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Consider the trajectory of …

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Terminal velocity in a magnetic field

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,

  1. Find the velocty v(t) of
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Missing energy in a rope and a capacitor

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.

  1. Find the force exerted on the end of the rope as function of the height y.
  2. 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 …

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Sliding on a block with a circular cut

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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  1. The position
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