Question

Why is no force act on a electron that have the same direction with magnetic field?Please explain.

Answer 1

Lorentz's force is the force acting over a charge that is moving in an electric and a magnetic field. Its formula is:

\[\bar F=q \left( \bar E+\bar v \times \bar B \right)\] If ther is no electric field, the formula can be reduced to:
\[\bar F=q \bar v \times \bar B\]The force is a result of the cross product of velocity (v) and magnetic field (B) times the charge. As you know, the cross product of two vectors is another vector whose module is given by the product of modules times the sin() of their angle. In this case:
\[\left| \bar F \right|=\left| q \right|\left| \bar v \right|\left| \bar B \right|\sin(\theta)\]If both velocity and magnetic field have the same direction, then \[\sin(\theta)=\sin(0)=0\rightarrow \left| \bar F \right|=0\]

Answer 2

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Answer 3

Thanks for the answer, why not cosine why sin?
And the electron will induce magnetic field the magnetic field is 90degree relative to external magnetic field but at 1 point there is no 2 magnetic field exist so we take the vector sum of these 2 magnetic field, since we take the vector sum that's mean there is an imbalance of magnetic field around the electron so the electron will be "push" to the lower density magnetic field ,but in formula the "push" force is zero. Why?
Is it because the force between the electron will cancel out after vector sum of 2 magnetic field(Induce by electron and external)?