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 states that the line integral of the magnetic field \( \vec{B} \) around any closed loop is equal to \( \mu_0 \) times the net current \( I_{\text{enc}} \) enclosed by the loop: \[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \] Where:

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

This law is analogous to Gauss’s law in electrostatics and is applicable in highly symmetric situations (e.g., long straight wires, solenoids). 

(II) Magnetic Field Outside a Long Solenoid:
As the solenoid becomes longer, the magnetic field lines inside become more uniform and denser, while the field lines outside begin to cancel due to opposite currents in adjacent turns.

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

Reason: The field lines from each turn outside the solenoid point in different directions and tend to cancel each other out due to symmetry. Hence, as length increases, the external field weakens and tends toward zero. 

(III) Flexible Loop Becoming Circular in Magnetic Field:
A current-carrying loop placed in an external magnetic field experiences a force that tends to minimize its potential energy. The magnetic pressure acts along the wire, pulling it into a shape that encloses maximum area for minimum perimeter — a circle. 
Reason: According to Lenz's law and the tendency to minimize magnetic potential energy, the system favors a configuration with maximum magnetic flux linkage — which occurs when the loop is circular. Thus, a flexible irregular loop deforms into a circle.