Problem 26 - Master Resource Book in JEE Main Physics

Question

A current of \(i\) ampere flows along an infinitely long straight thin walled tube, then the magnetic induction at any point inside the tube is (a) infinite (b) zero (c) \(\frac{\mu_{0} 2 i}{4 \pi r} \mathrm{~T}\) (d) \(\frac{\mu_{0} i}{2 r} \mathrm{~T}\)

Step-by-Step Solution

Answer

The magnetic induction inside the tube is zero; the answer is (b).

Step 1: Recall Ampère's Law

Ampère's Law states that the line integral of the magnetic field \( \mathbf{B} \) around a closed path is \( \mu_0 \) times the total current \( i \) passing through the enclosed area. Mathematically, it is written as \( \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 i \).

Step 2: Analyze the Geometry

The problem involves an infinitely long straight thin-walled tube with current \( i \) flowing along it. Because it is thin-walled and infinite, the tube behaves similarly to a hollow conductor with no current flowing through any enclosed area inside the inner surface of the tube.

Step 3: Apply Ampère's Law to a Path Inside the Tube

Consider a circular path inside the tube along which you apply Ampère's Law. Since there is no current enclosed by this path (because the tube is hollow and the current flows on its surface), the integral of the magnetic field \( \mathbf{B} \) along any path inside the tube would equal zero.

Step 4: Conclude the Solution

From Step 3, we conclude that the magnetic induction inside a current-carrying hollow tube is zero, since there is no enclosed current. This leads us to the selection of option \((b)\) as the correct answer.

Key Concepts

Magnetic Induction

Magnetic induction, also known as the magnetic field strength, is a fundamental concept in physics that describes the magnetic influence exerted by electric currents or magnetic materials. When a current flows through a conductor, it creates a magnetic field around it. The magnetic field can be visualized with magnetic field lines that form loops around the wire or conductor.

Key aspects of magnetic induction include:

• The strength of the field decreases with distance from the source of the magnetic field.
• The direction of the magnetic field lines follows the right-hand rule: if you point your thumb in the direction of the current, your fingers curl in the direction of the magnetic field.
• Magnetic induction is typically measured in tesla ( T ), a unit that reflects the intensity of the magnetic field.

This concept is essential for applying Ampère’s Law, which helps calculate the magnetic field around a given current. In this particular problem, magnetic induction inside an infinitely long hollow conductor is examined.

Thin-Walled Tube

A thin-walled tube is a special type of conductor in electromagnetism which has a very thin wall compared to its diameter. The material of the tube can carry an electric current along its surface. However, due to the nature of its construction, no current flows inside the hollow part of the tube.

This configuration is important when analyzing magnetic fields, because:

• The current is confined to the outer surface, and does not penetrate into the hollow center.
• Magnetic fields generated by the current on the surface do not affect the interior of the tube.
• This type of conductor is often studied as it simplifies the calculations, showing primary effects like the exclusion of magnetic field lines inside the tube.

In the context of the given exercise, the tube acts as a boundary for current, thus influencing the magnetic field behavior around and within the tube.

Hollow Conductor

A hollow conductor, in electrical and magnetic applications, refers to a conductor that is empty inside. An important phenomenon associated with hollow conductors is the absence of electric fields inside the hollow part, a concept exploited under certain conditions like in Faraday cages.

For magnetic fields, similar principles apply:

• In a hollow conductor carrying current only on its surface, the magnetic field inside should be zero.
• Ampère’s Law can be applied to explain this behavior: The magnetic field inside the hollow part is zero because there is no current enclosed in the loop path used in the calculations.
• This helps understand systems where magnetic shielding or field exclusion is necessary.

In the specific exercise, the key conclusion is that the magnetic induction at any point inside the hollow tube is zero, a direct application of the properties of a hollow conductor.