Consider an electron tube and the problem of determining the motion of electron in the tube. We can solve this problem using conservation of energy. Using conversation of energy we can derive the relation between potential and velocity of electron at any given point.

For this consider two points p1 and p2 within the tube as shown in the figure below.

Let v1 and V1 be velocity and potential of the electron at point p1 and let v2 and V2 be the velocity and potential at point p2.

From the conversation of energy, the energy of electron at point p1 and point p2 must be equal.

KE + PE at point p1 = KE + PE at point p2

where KE is kinetic energy and PE is the potential energy.

1/2*mv12+qV1 = 1/2*mv22+qV2

Consider that the point p1 is the point where the electron starts off so that it's velocity is zero. Also if this point is taken as the reference for potential then V1 is also 0. Then the above equation can written as,

1/2*mv22+qV2
or,
v2 = (-2QV2/m)1/2
or dropping the subscript for arbitrary point,

 v =  (-2QV/m)1/2

 This is a useful relation to determine the motion of charged particles such as electron in vacuum electron tube.


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