A circular current loop of radius R is in the nonuniform magnetic field of a bar

Question

A circular current loop of radius R is in the nonuniform magnetic field of a bar magnet. The current loop is placed as shown in the figure, with its center on the axis of the magnet and its plane perpendicular to it. The magnetic field at the position of the current loop makes an angle ? with respect to the magnet axis. The nonuniform magnetic field exerts a net force on the current loop.

(a) [5 points] Find an expression for the net magnetic force on the current loop.
(b) [5 points] Calculate the force if R=1.9 cm, I=0.55 A, B=260 mT, and ?=12°.

A circular current loop of radius R is in the nonuniform magnetic field of a bar magnet. The current loop is placed as shown in the figure, with its center on the axis of the magnet and its plane perpendicular to it. The magnetic field at the position of the current loop makes an angle with respect to the magnet axis. The nonuniform magnetic field exerts a net force on the current loop. net 2R (a) [5 points] Find an expression for the net magnetic force on the current loop (b) [5 points] Calculate the force if R=1.9 cm, 1=0.55 A, B=260 mT, and =129.

Explanation / Answer

a) The expression you used, namely
F = i*L*B*sin
Is valid only for a straight segment (with length L ) of a current carrying
conductor. Since the conducting segment lies along a straight line and
the direction of B is fixed, the angle between L and B is also fixed

In the case of conducting loop with radius R, the straight segments of the
current loop having infinitesimal lengths lie along the line tangent to the
curve of the loop. Those tangent lines cntinually change their directions
so that the angle between L and B is no longer constant but continually
changes too, so that sin is no longer constant. The net effect of the force
in (1) acting on all the infinitesimal straight segments of the conducting loop
is to exert a torque on the loop about an axis of symmetry of the loop that
also lies on the plane of the loop (this torque on a current loop can be more
easily understood by considering a rectangular loop with four straight line
segments only, each one having finite length). Be that as it may, the torque
on the current loop due to the magnetic field B producing the force is
= *B*cos
where = i*A = i**(R^2) is the magnetic moment of the conducting loop
perpendicular to the plane of the loop, and is the angle between and B.
Therefore, (2) can be rewritten as
= [ i**(R^2) ]*B*cos
In terms of the magnetic force itself, we have = F*R, so that
i**(R^2) ]*B*cos = F*R
Solving for F, we get
F = *i*R*B*cos

b) F = *i*R*B*cos =3.14* 0.55* 0.019*260e-3*cos12 =8.345e-3 N

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