Question

(I) Write Ampere’s circuital law in mathematical form and explain the terms used.
(II) As the current-carrying solenoid is made longer, the magnetic field produced outside it approaches zero. Why?
(III) A flexible loop of irregular shape carrying current, when located in an external magnetic field, changes to a circular shape. Give reason.

Hint:

- Ampere’s law is powerful for calculating magnetic fields in symmetric cases. - Long solenoids are ideal field generators with negligible external field. - Magnetic forces tend to reshape current-carrying loops to minimize energy — hence circular shapes are preferred in magnetic fields.

Answer 1

(I) Ampere’s Circuital Law:
Ampere’s circuital law posits that the line integral of the magnetic field \( \vec{B} \) around any closed loop equals \( \mu_0 \) multiplied by the net current \( I_{\text{enc}} \) enclosed by the loop: \[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \] Where:

• \( \vec{B} \) denotes the magnetic field vector
• \( d\vec{l} \) represents an infinitesimal vector element of the closed path
• \( \mu_0 \) signifies the permeability of free space
• \( I_{\text{enc}} \) is the total current contained within the loop

This law is analogous to Gauss’s law in electrostatics and is applicable in scenarios with high symmetry, such as long straight wires or solenoids.

(II) Magnetic Field Outside a Long Solenoid:
As a solenoid's length increases, the magnetic field lines within it become more uniform and denser. Conversely, the field lines outside tend to cancel due to opposing currents in adjacent turns.

In the theoretical case of an infinitely long solenoid, the external magnetic field is zero: \[ B_{\text{outside}} = 0 \]

Reason: Outside the solenoid, the field lines from each turn are oriented in different directions and effectively cancel each other due to symmetry. Consequently, as the solenoid's length grows, the external field diminishes, approaching zero. 

(III) Flexible Loop Becoming Circular in Magnetic Field:
A current-carrying loop within an external magnetic field experiences a force that drives it toward minimizing its potential energy. The magnetic pressure acts along the wire, compelling it to adopt a shape that maximizes the enclosed area for a given perimeter — a circle. 
Reason: In accordance with Lenz's law and the principle of minimizing magnetic potential energy, the system favors a configuration with maximum magnetic flux linkage, which is achieved when the loop is circular. Therefore, an irregularly shaped flexible loop will deform into a circle.