A conducting circular loop of radius a carries a constant...

A conducting circular loop of radius a carries a constant current I. It is placed in a uniform magnetic field $\mathrm{B}^{-}$, such that $\mathrm{B}^{-}$ is perpendicular to the plane of the Loop. The magnetic force acting on the Loop is (a) $\mathrm{B}^{-} \operatorname{Ir}$ (b) $\mathrm{B}^{-} \mathrm{I} \pi \mathrm{r}^{2}$ (c) Zero (d) BI $(2 \pi \mathrm{r})$

A conducting circular loop of radius a carries a constant
current I. It is placed in a uniform magnetic field $\mathrm{B}^{-}$, such
that $\mathrm{B}^{-}$ is perpendicular to the plane of the Loop. The magnetic force acting on the Loop is
(a) $\mathrm{B}^{-} \operatorname{Ir}$
(b) $\mathrm{B}^{-} \mathrm{I} \pi \mathrm{r}^{2}$
(c) Zero
(d) BI $(2 \pi \mathrm{r})$

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field, which, in the context of a current-carrying conductor, is the integral of the magnetic force on the moving charges comprising the current. In differential form, an element of current-carrying conductor experiences a force proportional to the cross product of the current element and the magnetic field, highlighting that the direction and magnitude of the force depend on both the orientation of the current element and the magnetic field.

Uniform Magnetic Field

A uniform magnetic field is one that has the same magnitude and direction at every point in space. This uniformity means that any spatial variation in the magnetic forces acting on symmetric structures, such as a circular current loop, cancels out, leading to no net force even though individual segments of the loop experience forces.

Net Force on a Current Loop

For a current loop placed in a uniform magnetic field, although different parts of the loop experience individual magnetic forces due to the Lorentz force, the symmetry of the loop ensures that these forces cancel out. This cancellation results in no net translational force acting on the loop, even though a torque might be present if the loop’s magnetic dipole moment is not aligned with the field.

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity associated with the current loop, defined by the product of the current and the area of the loop, with a direction given by the right-hand rule. It characterizes the overall magnetic behavior of the loop in external fields and is responsible for phenomena such as experiencing a torque in non-uniform fields, though in a uniform field its contribution to net force is null.

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