Magnetic Field

Electricity and Magnetism Level 3

A fixed thin long wire carries a current $I_0$ . A small particle of mass $m$ and charge $q$ is at rest at a distance of $x_0$ from the wire. The particle is now projected with a velocity $v_0$ towards the wire and perpendicular to it. Find the minimum separation between the particle and the wire.

x_0

\tan (\frac{\pi }{12})

x_0

In(\frac{4\mu _0qI_0}{\pi mv_0})

x_0

e^{-2\pi mv_0/q\mu _0I_0}

x_0^2

(1-\frac{mv_0}{q\mu _0I_0})

1 solution

Mat Baluch
Aug 19, 2015

One can solve the equation of motions and derive the correct answer.

Here i propose a different approach: determine the correct answer from the 4 propositions by elimination, as if we were trying to solve as fast as possible the MCQ. I recommand this exercise as it is very usefull in a competition situation.

First, let us understand the physics of the problem. The particle is projected perpendicularly to the wire, which results by Lorrentz formula $F = qv \times B$ , and since the field lines are circles perpendicular to the wire (ideal case of infinitly long wire used here), in a vertical force on the particle . This force is upward or downward, depending on the signes of $q$ and $I_{0}$ , but with no influence on the final separation, by symmetry.

As the particle is deflected upward (or downward), a vertical component of speed appears, creating of radial component of Lorrentz force, pushing the particle away from the wire. We are guaranteed that the repulsing effect will eventually overcome inertia, as the magnetic field tends to infinity close to the wire.

After this brief analysis, we can conclude:

the minimum separation is defined for any parameter, and lies between $0$ and $x_{0}$
for $v_{0} = 0$ this should be $x_{0}$ , as the particle will not move
for a given $v_{0}$ , this separation should decrease when $I_{0}$ increases, as the field $B$ increases with $I_{0}$ , and so does the repulsion

This shows that only choice n°2 can be correct : answer 1 is not homogeneous to a length, answer 3 tends to infinity when $v_{0} = 0$ (sic !), and answer 4 does not depends on $v_{0}$

Magnetic Field

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