Magnetic Jack

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Jack is given to construct two externally tangent circles in his geometry exam. However, he has no idea how to use his geometry tools, so instead he thinks of another plan.

He places a charged particle \(P\) having mass \(m= 0.3 \text{ kg}\) and charge \(q= 2 \text{ Coloumbs} \) at a point \(A\) on the paper. Jack gives it a push, and the particle starts moving at a constant velocity \(v= 3 \text{ cm/s}\). Instantaneously after pushing the particle, Jack applies a magnetic field \(B= 6 \text{ Teslas}\) perpendicular to the paper, and the particle changes its trajectory. Once the particle returns to its original position after completing a full revolution, Jack turns the magnetic field off.

Now, Jack breaks the particle into two parts, each having half the mass and half the charge of the original particle. Jack places one half at another point \(B\) on the paper, and does the same experiment.

It turns out that when the trajectories of both the particles are combined, they trace out a pair of externally tangent circles, exactly what Jack wanted! Let \(L\) be the distance between \(A\) and \(B\) which makes this possible. Find \(40L\) in centimeters.

Details and assumptions

  • The coefficient of friction of the paper is zero.
  • Don't try this experiment in your mathematics exam!
  • When performing the experiment for the second time, Jack uses the same magnetic field \(\left ( 6 \text{ Teslas} \right ) \) and pushes the particle giving it the same constant velocity \( \left ( 3 \text{ cm/s} \right ) \).
Image source: Edudemic

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