Why Does a Magnetic Field Do Zero Work?

[conorordan]

Homework Statement

This isn't strictly homework, I'm going over my lecturer's notes before the exam and I don't understand a part of this proof for magnetic fields doing no work.

This is the second proof he gave, the first being the more standard $\vec{B}\cdot\textit{d}\vec{l}=0$

Homework Equations

So in the second proof he does it like this;

I cannot figure out where he gets the factor of a half in the third line, to make the equation come out as change in kinetic energy = 0.

 

[tiny-tim]

hi conorordan!

I cannot figure out where he gets the factor of a half in the third line

product rule …

d(v·v)/dt = v·dv/dt + dv/dt·v = 2 v·dv/dt