Worked exercise — rectangular loop in a uniform field

Problem

A rectangular loop with sides $a = 2\;\text{m}$ and $b = 0{,}5\;\text{m}$ carries a current $i = 2\;\text{A}$ and is immersed in a field $\vec{B}$ pointing into the page, of magnitude $|\vec{B}| = 2\;\text{T}$. The long sides are horizontal. Calculate the forces on the four sides.

Solution

I use the Laplace force $\vec{F} = i\,\vec{\ell}\times\vec{B}$ on each side.

Long sides ($a = 2\;\text{m}$), perpendicular to $\vec{B}$: $|F_1| = i\,a\,|\vec{B}|\,\sin 90^\circ = 2\cdot 2\cdot 2 = 8\;\text{N}$ on each, but in opposite senses (one upwards, the other downwards): they cancel out.

Short sides ($b = 0{,}5\;\text{m}$), also perpendicular to $\vec{B}$: $|F_2| = i\,b\,|\vec{B}| = 2\cdot 0{,}5\cdot 2 = 2\;\text{N}$ also in opposite senses: they cancel out.

$\ev{\vec{F}_\text{resultant} = 0}$

The loop does not translate. But the forces $F_1$ and $F_1'$ on the long sides, equal and opposite but applied along different lines, form a couple that tends to rotate the loop. This is the principle of the electric motor: zero resultant, non-zero torque.

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Topics: Magnetismo Concepts: Forza di Laplace · Dipolo magnetico · Momento di una forza