The Definition of B — Question 1

Question 1

A magnetic cross product fixes both the force components and their perpendicularity

A particle of charge $q=2.00$ microcoulomb has velocity $(3\hat x+4\hat y)10^5$ m/s in a uniform field $.200\hat z$ tesla. No other force acts.

1. Find the magnetic-force vector and its magnitude.

2. Calculate the instantaneous power and explain the consequence for kinetic energy.

Question 1 - Solution

A magnetic cross product fixes both the force components and their perpendicularity

(a) Method and calculation

$$\vec F=q\vec v\times\vec B=qB(v_y\hat x-v_x\hat y).$$$$\vec F=(.160\hat x-.120\hat y)\,\mathrm N,\quad F=.200\,\mathrm N.$$

(b) Interpretation and further calculation

$$P=\vec F\cdot\vec v=(.160)(3\times10^5)-(.120)(4\times10^5)=0.$$$$\frac{dK}{dt}=0\Rightarrow K=\text{constant}.$$

Physical check

The force changes the direction of velocity, not its magnitude, in a purely magnetic field.