Problem 62 - Physics

Question

Refer to Interactive Solution \(21.6221 .62\) at for help with problems like this one. A very long, hollow cylinder is formed by rolling up a thin sheet of copper. Electric charges flow along the copper sheet parallel to the axis of the cylinder. The arrangement is, in effect, a hollow tube of current \(I\). Use Ampère's law to show that the magnetic field (a) is \(\mu_{0} I /(2 \pi r)\) outside the cylinder at a distance \(r\) from the axis and (b) is zero at any point within the hollow interior of the cylinder. (Hint: For closed paths, use circles perpendicular to and centered on the axis of the cylinder.)

Step-by-Step Solution

Answer

Outside the cylinder, \( B = \frac{\mu_0 I}{2\pi r} \); inside, \( B = 0 \).

Step 1: Understanding Ampère's Law

Ampère's Law states: \( \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enclosed} \), where \( \mathbf{B} \) is the magnetic field, \( d\mathbf{l} \) is the differential length element of a closed loop, and \( I_{enclosed} \) is the current enclosed by the loop.

Step 2: Applying Ampère's Law outside the cylinder

For a point outside the cylinder, consider an Amperian loop that is a circle of radius \( r \) centered on the axis of the cylinder. The symmetry implies that \( \mathbf{B} \) is constant in magnitude and tangential around the loop, leading to \( \oint B \cdot dl = B(2\pi r) \). The enclosed current \( I_{enclosed} = I \) is the total current through the cylinder.

Step 3: Solving for the magnetic field outside

From Ampère's Law, we have the equation \( B(2\pi r) = \mu_0 I \). Solving for \( B \), we obtain \( B = \frac{\mu_0 I}{2\pi r} \), showing that the magnetic field outside the cylinder at a distance \( r \) is \( \mu_0 I /(2 \pi r) \).

Step 4: Applying Ampère's Law inside the cylinder

For a point inside the hollow part of the cylinder, consider an Amperian loop again centered on the axis, but with radius smaller than that of the cylinder wall. No current flows through any surface bounded by this loop, so \( I_{enclosed} = 0 \).

Step 5: Solving for the magnetic field inside

With \( I_{enclosed} = 0 \), Ampère's Law becomes \( B(2\pi r) = 0 \). Thus, the magnetic field \( B \) is zero within the hollow interior of the cylinder, as expected.

Key Concepts

Magnetic Field

The concept of a magnetic field is central to understanding magnetism and electromagnetism. A magnetic field is a vector field surrounding a magnet or an electric current, indicating the force that would act on a magnetic pole or moving charge placed within it. This field is represented by lines that illustrate the direction of magnetic forces. The closer these lines are, the stronger the magnetic field. It is a fundamental aspect of understanding how electrical currents interact with the space around them.

In this exercise, we specifically look at the magnetic field around a hollow cylinder carrying an electric current. Using Ampère's Law, we find that the magnetic field outside the cylinder is given by the formula \( B = \frac{\mu_0 I}{2\pi r} \). This formula tells us that the strength of the magnetic field decreases inversely with distance from the cylinder, meaning the further away you are, the weaker the field. Conversely, Ampère's Law reveals that there is no magnetic field inside the hollow part of the cylinder, showcasing the fascinating distribution of magnetic forces.

Hollow Cylinder

A hollow cylinder, in this context, refers to a geometrical shape formed by rolling a thin sheet of conductive material, like copper, such that there is a cylindrical void through the center. In our case, this cylinder is very long, which simplifies calculations because the ends do not affect the results.

When considering magnetic fields around a hollow cylinder carrying current, understanding its geometry helps us apply Ampère's Law effectively. The currents flow along the sheet parallel to the axis, forming a hollow tube of current. This configuration influences how the magnetic field behaves around and inside the cylinder. With the current distribution being uniform along the cylinder's wall, it ensures that there is circular symmetry around the axis. These symmetries are what allow us to make calculations using simpler circular Amperian loops perfectly centered on the axis, giving clear results based on the radial distance from the center.

Electric Current

Electric current is the flow of electric charge and is the source of magnetic fields in conductors. In our problem, the electric current flows along a conductive copper sheet forming a hollow cylinder. This current is represented by \( I \), the total amount of charge flowing per unit time, which creates a magnetic field around the conductor.

The direction and magnitude of this current determine the characteristics of the magnetic field. The current's motion along the cylinder walls leads to a magnetic field organized in concentric circles around the cylinder. The computation of these fields, as done using Ampère's Law, relies on accurately factoring in the total current passing the chosen pathway. By focusing on a cylindrical configuration, it allows different Amperian paths to be examined, elucidating the field behavior both outside and within the hollow part of the cylinder.

Magnetism

Magnetism is a fundamental force, influencing how materials respond to magnetic fields. Electric currents generate magnetic fields, a principle exploited in various applications, from electric motors to electromagnetic shielding. In our exercise, understanding how magnetism operates through Ampère's Law is essential for predicting the magnetic field configuration inside and outside the hollow copper cylinder.

Magnetism does not exist in isolation within the hollow part of the cylinder due to the absence of an enclosed current, which results in a zero magnetic field there. This zero-field region demonstrates a significant aspect of magnetic shielding, where fields can be absent within specific structures. Outside the cylinder, the magnetic field finds its strength proportional to the current and inversely proportional to the distance from the cylinder. This application shows how magnetism and electromagnetism principles can be used in designing and interpreting the behavior of magnetic fields in complex geometries.