Question

Is it really possible that when electron beams enters into a uniform magnetic field perpendicularly, it only changes only it's path but it velocity doesn't decrease.why? If we observe the nature we find that there must be decrease in velocity if anything changes it path due to the force of other?

Answer 1

the speed of electron remains same but its velocity changes.

Answer 2

The speed stays the same because the force (and therefore the acceleration) is perpendicular to the velocity. You can look at it analytically ->
\[ \vec{F} = m\frac{d\vec{v}}{dt} = q(\vec{v}\times\vec{B})\]
\[\vec{v}\cdot \frac{d\vec{v}}{dt} = \frac{d}{dt}(\vec{v}\cdot \vec{v}) - \frac{d\vec{v}}{dt}\cdot \vec{v} = \frac{d}{dt}v^2 - \frac{q}{m}(\vec{v}\times \vec{B})\cdot \vec{v} = \vec{v}\cdot \frac{q}{m}(\vec{v}\times \vec{B})\]
therefore,
\[\frac{d}{dt}v^2 = \frac{2q}{m} \vec{v} \cdot (\vec{v}\times \vec{B}) = \frac{2q}{m}\vec{B}\cdot (\vec{v}\times \vec{v}) \]
but
\[\vec{v}\times \vec{v} = 0\]
so
\[\frac{d}{dt} v^2 = 0 \rightarrow v^2 = constant\]
and therefore v itself is constant, v being the magnitude of the velocity.

Answer 3

Correct, although I don`t really understand these equations, only the first line. Velocity does not change because the magnetic force is always perpendicular to it. Good question, now you can see an important difference between magnetic and electric field. E field can change the magnitude (that means, the kinetic energy) and direction of the velocity, but B field can only change the direction.