A helium nucleus makes a full rotation in a circle of radius 0.8 m in two seconds. The value of

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A helium nucleus makes a full rotation in a circle of radius...

A helium nucleus makes a full rotation in a circle of radius $0.8 \mathrm{~m}$ in two seconds. The value of the magnetic field $B$ at centre of circle will be :
(a) $\frac{10^{-19}}{\mu_{0}}$
(b) $10^{-19} \mu_{0}$
(c) $2 \times 10^{-19} \mu_{0}$
(d) $\frac{2 \times 10^{19}}{\mu_{0}}$

Key Concepts

Effective Current from a Moving Charge

A moving charge in a circular path can be modeled as an effective current, defined as I = q/T, where q is the charge and T is the period of rotation. This approach transforms a discrete charge moving periodically into an equivalent continuous current loop, allowing the application of formulas such as the one for the magnetic field at the center of a current loop.

Circular Motion and Period

Circular motion refers to the movement of an object along a circular path, characterized by parameters such as radius, period, and frequency. The period is the time taken to complete one full revolution. This measure is directly used to determine the effective current of a moving charge in a circular orbit, making it a critical element in problems involving magnetic fields generated by moving charges.

Magnetic Field of a Current Loop

A current loop produces a magnetic field that is strongest at the center of the loop. The field is typically calculated using formulas derived from the Biot–Savart law, one of which gives the magnetic field at the center of a circular loop as B = ??I/(2R), where I is the current and R is the radius of the loop. This concept is fundamental in understanding how moving charges give rise to magnetic effects, and it is broadly applicable in electromagnetism.

Charge of the Particle

In problems involving magnetic fields due to moving charges, it is important to know the charge of the particle. For instance, a helium nucleus, which typically consists of two protons and thereby carries a charge of +2e, contributes to the calculation of the effective current when it completes a circular orbit. Understanding the particle’s charge is essential for accurately determining the magnetic effects it produces.

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