# Magnetic field in asymmetric hollow cylinder

• bobred
In summary, the problem involves a cylinder with a cylindrical hole that is displaced in the x-direction. The current density is represented by \textbf{J}=J_z\textbf{e}_z and the goal is to show that a uniform magnetic field exists inside the hole, given by \textbf{B}=\frac{\mu_0}{2}J_zd\textbf{e}_y. The solution involves using the equation for the magnetic field inside a whole cylinder and adding the contribution from the cylindrical hole, which can be modeled as a charge density of -J_z\textbf{e}_z. However, a mistake was made in the calculation and the correct approach is to use \textbf{B
bobred

## Homework Statement

Cylinder of radius $a$ and a cylindrical hole $b < a$ is displaced a distance $d$ in x-direction. Current density $\textbf{J}=J_z\textbf{e}_z$. Show that a uniform magnetic field inside the hole is

$\textbf{B}=\frac{\mu_0}{2}J_zd\textbf{e}_y$

## Homework Equations

Using previous result of whole cylinder

$\textbf{B}=\frac{\mu_0}{2}J_z(-y\textbf{e}_x + x\textbf{e}_y)$

and that the cylindrical hole can be modeled as a charge density of $-J_z\textbf{e}_z$.

## The Attempt at a Solution

I tried superposition of fields so

$\textbf{B}_1=\frac{\mu_0}{2}J_z(-y\textbf{e}_x + x\textbf{e}_y)$

$\textbf{B}_2=-\frac{\mu_0}{2}J_z(-y\textbf{e}_x + (x + d)\textbf{e}_y)$

$\textbf{B}= \textbf{B}_1 + \textbf{B}_2$ to which I get

$\textbf{B}=-\frac{\mu_0}{2}J_zd\textbf{e}_y$

Any ideas, is this the right approach?

Sorted, silly mistake

$\textbf{B}_2=-\frac{\mu_0}{2}J_z(-y\textbf{e}_x + (x + d)\textbf{e}_y)$

should be

$\textbf{B}_2=-\frac{\mu_0}{2}J_z(-y\textbf{e}_x + (x - d)\textbf{e}_y)$

## 1. What is a magnetic field in an asymmetric hollow cylinder?

A magnetic field in an asymmetric hollow cylinder is a region in space where a magnetic force can be detected. This force is caused by the movement of electrically charged particles within the cylinder.

## 2. How is a magnetic field created in an asymmetric hollow cylinder?

A magnetic field in an asymmetric hollow cylinder is created by the flow of current through the cylinder's walls. This current generates a circular magnetic field around the cylinder.

## 3. What factors affect the strength of the magnetic field in an asymmetric hollow cylinder?

The strength of the magnetic field in an asymmetric hollow cylinder is affected by the amount of current flowing through the cylinder, the shape and size of the cylinder, and the material of the cylinder's walls.

## 4. How is the direction of the magnetic field determined in an asymmetric hollow cylinder?

The direction of the magnetic field in an asymmetric hollow cylinder is determined by the right-hand rule, which states that if you point your right thumb in the direction of the current flow, your fingers will wrap around the cylinder in the direction of the magnetic field.

## 5. What are some real-world applications of magnetic fields in asymmetric hollow cylinders?

Magnetic fields in asymmetric hollow cylinders have various applications, such as in electric motors, generators, and magnetic resonance imaging (MRI) machines. They are also used in particle accelerators and magnetic levitation trains.

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