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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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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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


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

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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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