The Definition of B — Question 7

Question 7

A curved trajectory can have nonzero magnetic force and exactly zero magnetic work

A nonrelativistic particle of mass m and charge q follows an arbitrary path in a stationary magnetic field, with no other force. Its initial speed is v0.

1. Prove that its kinetic energy and speed remain constant even if the field varies with position.

2. If its velocity changes from $v_0\hat x$ to $v_0\hat y$, find the momentum-change magnitude and explain why this does not contradict zero work.

Question 7 - Solution

A curved trajectory can have nonzero magnetic force and exactly zero magnetic work

(a) Method and calculation

$$\frac{dK}{dt}=m\dot{\vec v}\cdot\vec v=q(\vec v\times\vec B)\cdot\vec v=0.$$$$K=\tfrac12mv_0^2,\quad |\vec v|=v_0,\quad W_B=\int\vec F_B\cdot d\vec r=0.$$

(b) Interpretation and further calculation

$$\Delta\vec p=mv_0(\hat y-\hat x),\quad|\Delta\vec p|=\sqrt2mv_0.$$$$\Delta K=\frac{p_f^2-p_i^2}{2m}=0.$$

Physical check

Work depends on changes in momentum magnitude squared, whereas impulse depends on the full vector change.