Mathematics

Problem

A technician wishes to maintain a constant mechanical force of

1.2\,\text{N}

on a straight aluminium rod

0.25\,\text{m}

long placed in a uniform magnetic field of

0.30\,\text{T}

while the rod is gradually rotated in the field. a. (i) Starting from the relevant expression for the magnetic force on a current-carrying conductor, derive an equation giving the current

I

required to keep the force constant at

1.2\,\text{N}

as a function of the angle

\theta

between the rod and the magnetic field. b. (ii) Using your derived equation, calculate the current needed when

\theta = 60^{\circ}

and when

\theta = 30^{\circ}

. c. (iii) Comment on how the current must be varied as the rod is rotated from a position perpendicular to the field to a position parallel to the field.

Intuitive explanation

The key here is understanding how the magnetic force on a current-carrying conductor depends on the angle between the rod and the field. The force formula $F = BIL\sin\theta$ shows that the magnetic force varies with orientation—when the rod is perpendicular to the field, you get maximum force, but as it rotates, the effective component decreases. To keep the total force constant at $1.2\,\text{N}$ , you'll need to adjust the current as the angle changes, using the inverse relationship: as $\sin\theta$ drops, the current $I$ must rise to compensate.

Intuitive explanation

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