11.4 Magnetic force on a current-carrying conductor  (Page 2/3)

The magnetic force on any single charge carrier is $e{\overrightarrow{v}}_{\text{d}}\times \overrightarrow{B},$ so the total magnetic force $d\overrightarrow{F}$ on the $nA·dl$ charge carriers in the section of wire is

We can define dl to be a vector of length dl pointing along ${\overrightarrow{v}}_{\text{d}},$ which allows us to rewrite this equation as

or

This is the magnetic force on the section of wire. Note that it is actually the net force exerted by the field on the charge carriers themselves. The direction of this force is given by RHR-1, where you point your fingers in the direction of the current and curl them toward the field. Your thumb then points in the direction of the force.

To determine the magnetic force $\overrightarrow{F}$ on a wire of arbitrary length and shape, we must integrate [link] over the entire wire. If the wire section happens to be straight and B is uniform, the equation differentials become absolute quantities, giving us

This is the force on a straight, current-carrying wire in a uniform magnetic field.

Balancing the gravitational and magnetic forces on a current-carrying wire

A wire of length 50 cm and mass 10 g is suspended in a horizontal plane by a pair of flexible leads ( [link] ). The wire is then subjected to a constant magnetic field of magnitude 0.50 T, which is directed as shown. What are the magnitude and direction of the current in the wire needed to remove the tension in the supporting leads?

Strategy

From the free-body diagram in the figure, the tensions in the supporting leads go to zero when the gravitational and magnetic forces balance each other. Using the RHR-1, we find that the magnetic force points up. We can then determine the current I by equating the two forces.

Solution

Equate the two forces of weight and magnetic force on the wire:

$mg=IlB.$

Thus,

$I=\frac{mg}{lB}=\frac{(0.010\text{kg})(9.8{\text{m/s}}^2)}{(0.50\text{m})(0.50\text{T})}=0.39\text{A.}$

Significance

This large magnetic field creates a significant force on a length of wire to counteract the weight of the wire.

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Calculating magnetic force on a current-carrying wire

Strategy