Ampere’s law and symmetry

Problem 3.231 from Irodov.

A current I flows along a lengthy straight wire, as shown in the figure below. From the point O the current spreads radially all over an infinite conducting plane perpendicular to the wire. Find the magnetic field above and below the plane.

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A wire carrying current to an infinite conducting plane. We indicate the radial current on the plane with a few lines, but there is current flowing along every point of the plane.


We will use this problem to demonstrate the use of Ampere’s law, starting with some simple scenarios. Our goal is to understand …

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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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Electromagnetic induction in a twisted loop

JEE Advanced 2017 Paper 1, Question 5

A circular insulated copper wire loop is twisted to form two loops of area A and 2 A as shown in the figure. At the point of crossing the wires remain electrically insulated from each other. The entire loop lies in the plane (of the paper). A uniform magnetic field {\bf B} points into the plane of the paper. At t=0, the loop starts rotating about the common diameter as axis with a constant angular velocity \omega in the magnetic field. Which of the following options is/are correct?

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  1. The emf induced in the loop is proportional to the sum
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An infinite ladder of resistors

Problem 3.152 of Irodov

The figure below shows an inifite circuit formed by the repetition of the same link, consisting of resistance R_1 = 4.0 \ \Omega and R_2 = 3.0 \ \Omega. Find the resistance of this circuit between points A and B.

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Let’s denote the resistance between the points A and B by R_{AB}. Since the circuit is infinite, removing the first R_1 and R_2 resistors gives the same arrangement back again — the arrangement is self-similar. That means, the resistance between the points C and D is just R_{AB} without the left-most R_1 and R_2 resistors, and we may redraw the circuit as shown below.

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It is now straightforward to calculate the resistance,


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